sistemas de traslaciones: generadores de marcos y...
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UNIVERSIDAD DE BUENOS AIRES
Facultad de Ciencias Exactas y Naturales
Departamento de Matematica
Sistemas de traslaciones:generadores de marcos y aproximacion
Tesis presentada para optar al tıtulo de Doctora de la Universidad de Buenos Aires en elarea Ciencias Matematicas
Carolina Alejandra Mosquera
Director de tesis: Carlos Cabrelli
Consejero de estudios: Carlos Cabrelli
Buenos Aires, 2015.
Fecha de defensa: 16 de marzo de 2015.
Sistemas de traslaciones:generadores de marcos y aproximacion
(Resumen)
Los espacios invariantes por traslaciones son espacios cerrados de L2(Rd) que son invari-antes por traslaciones enteras. Estos espacios juegan un rol muy importante en teorıade aproximacion, analisis armonico, teorıa de wavelets, muestreo y procesamiento deimagenes.
Un conjunto de funciones Φ es un conjunto de generadores de un espacio invariante portraslaciones V si la clausura en L2(Rd) del espacio generado por las traslaciones enterasde las funciones de Φ coincide con V. Es importante saber cuando es posible obtener unconjunto minimal de generadores de un conjunto de generadores dado en Φ. Con refer-encia a esta pregunta, Bownik y Kaiblinger en 2006, mostraron que puede obtenerse de Φ
un conjunto minimal de generadores de V a traves de combinaciones lineales de sus ele-mentos. Como tomar combinaciones lineales de un numero finito de funciones preservapropiedades tales como la suavidad, el decaimiento, tener soporte compacto, ser de bandalimitada, etc., una consecuencia interesante del resultado de Bownik y Kaiblinger es quesi los generadores de V tienen una propiedad adicional, existe un conjunto minimal degeneradores que hereda esa propiedad.
En muchos problemas que involucran espacios invariantes por traslaciones, es impor-tante que el sistema de traslaciones posea una estructura analıtica particular como ser unabase ortonormal, una base de Riesz o un marco. Luego, resulta interesante saber cuandoun conjunto minimal de generadores obtenido a traves de combinaciones lineales de unodado preserva la misma estructura que el sistema de generadores original. En esta tesisdaremos respuesta a esta pregunta para el caso de marcos, bases de Riesz y bases ortonor-males.
Los espacios invariantes por traslaciones enteras pueden tener ademas extra invarian-cia, es decir, pueden ser invariantes bajo otras traslaciones ademas de las enteras. Estosespacios con extra invariancia son importantes en aplicaciones especialmente en aquellasdonde el error que se produce en ciertas aproximaciones es relevante.
iv
En esta tesis, dado un conjunto de datos F = f1, . . . , fm ⊆ L2(Rd), probamos laexistencia y mostramos como construir un “espacio invariante por traslaciones enteraspequeno” V mas cercano a F sobre cierta clase de subespacios cerrados de L2(Rd). Elespacio aproximante debe tener cierta extra-invariancia, es decir, debe ser invariante bajotraslaciones sobre un subgrupo aditivo de Rd que contenga estrictamente a Zd, ya fijado.Aquı con pequeno, nos referimos a que el subespacio solucion este generado por las trasla-ciones enteras de un pequeno numero de generadores. Ademas, damos una expresion delerror en funcion de los datos y construimos un marco de Parseval para el espacio optimo.
Por otro lado, consideramos el problema de aproximar F por espacios de Paley-Wienergeneralizados de Rd, que estan generados por las traslaciones enteras de un numero finitode funciones. Estos espacios pueden verse como espacios invariantes por traslacionesenteras finitamente generados que son Rd invariantes.
Ademas mostramos la relacion entre estos espacios y los conjuntos “multi-tile” de Rd
y la conexion con resultados recientes acerca de bases de Riesz de exponenciales.
Palabras claves: espacios invariantes por traslaciones enteras; marcos; bases de Riesz;bases ortonormales; muestreo; extra invariancia; espacios de Paley-Wiener.
Systems of translates:frame generators and approximation
(Abstract)
Shift invariant spaces are closed subspaces of L2(Rd) that are invariant under integer trans-lations. They play an important role in approximation theory, harmonic analysis, wavelettheory, sampling and signal processing.
A set of functions Φ is a set of generators for a shift invariant space V if the closureof the space spanned by all integer translations of the functions in Φ agrees with V. It isinteresting to know whether it is possible to obtain a minimal set of generators from thegiven generators in Φ. Concerning this question, Bownik and Kaiblinger in 2006, showedthat a minimal set of generators for V can be obtained from Φ by linear combinationsof its elements. Since linear combinations of a finite number of functions preserve prop-erties such as smoothness, compact support, bandlimitedness, decay, etc, an interestingconsequence of Bownik and Kaiblinger’s result is that if the generators for V have someadditional property, there exists a minimal set of generators that inherits this property.
In many problems involving shift invariant spaces, it is important that the system oftranslates bears a particular functional analytic structure such as being an orthonormalbasis, a Riesz basis or a frame. Therefore, it is interesting to know when a minimal setof generators obtained by taking linear combinations of the original one has the samestructure. In this thesis we answer this question completely, in the case of frames, Rieszbases and orthonormal bases.
Shift invariant spaces can also have extra invariance, that is they could be invariantunder translates other than integers. Such spaces are important in applications speciallyin those where the error in approximations is an issue.
In this thesis, given an arbitrary finite data F = f1, . . . , fm ⊆ L2(Rd), we prove the ex-istence and show how to construct a “small shift invariant space” V that is the “closest” tothe data F over certain class of closed subspaces of L2(Rd). The approximating subspaceis required to have extra-invariance properties, that is to be invariant under translationsby a prefixed additive subgroup of Rd containing Zd. Here small means that our solution
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subspace should be generated by the integer translates of small number of generators. Wegive an expression for the error in terms of the data and construct a Parseval frame for theoptimal space.
We also consider the problem of approximating F from generalized Paley-Wienerspaces of Rd, that are generated by the integer translates of finite number of functions.These spaces can be seen as finitely generated shift invariant spaces that are Rd invariant.
We show the relations between these spaces and multi-tile sets of Rd, and the connec-tions with recent results on Riesz basis of exponentials.
Key words: shift-invariant spaces; frames; Riesz bases; orthonormal bases; sampling;extra invariance; Paley-Wiener spaces.
Agradecimientos
A Carlos por su guıa, dedicacion, paciencia, confianza y respeto durante estos anos detrabajo en el doctorado.
A Ursula por conducir junto con Carlos el grupo de Analisis Real y Armonico que tantome ayudo en mi formacion.
A papa Mario, mama Lucıa, hermana Flor y abuelos por alentarme y acompanarmedurante esta etapa de mi vida. A los Del Pezzo: Sabri, Yami y Martın por incorporarmecon carino a su familia.
A Ani, Magui, Marian y Vicky por ser amigas de fierro. Por acompanarme, escucharmey aconsejarme en las buenas y en las malas durante estos anos! No se que hubiera hechosin ustedes. Las quiero!
A los ex y actuales companeros de la ofi: Magui, Vicky, Cris, Ani, Dany, Chris, Colo,Fer, Leo, Nico, Mazzi, Savra, Anto, Turco, Roman y Andre. Gracias por la linda cotid-ianeidad compartida, por las charlas y por construir mi mejor lugar de trabajo. Graciastambien por la yerba y agua “prestadas”!
A las chicas del oeste: Pau, Zule, Naty, Vale y Gi por la linda amistad de la infanciarecobrada en los ultimos anos.
A mis amigos y companeros de escuelas y congresos: Iso, Marce, Mari, Mauri, Maikel,Lucas, Cris y Marian. Por ser la mejor parte de los congresos y que sean por muchos viajesjuntos mas!
A la gente del dm por las innumerables charlas de pasillo, almuerzos, cafecitos y salidascompartidas: Mer, Sigrid, San, Nico S., Pau, Anita, Coty, Vendra, Muro, Dano, Colo,Laplagne, Rela, Manu, los chicos de la 2038, Gaby, Vero, Lau, Juli, etc.
A Vicky, Carlos, Lean y Julio porque fue un placer trabajar con ellos.
A Moni, Deby y Lili por estar ahı siempre que las necesite durante el doctorado y porlas lindas charlas compartidas.
A Julio por su generosidad y por ofrecerme trabajar con el, dandole ası un impulso ami vida academica.
viii Agradecimientos
A las chicas de paddle por los divertidos partidos de los martes!
A Lean por su amor. Por ser mi sosten y mi cable a tierra, por contenerme, escucharme,involucrarse y aconsejarme siempre. Por ser uno de los principales motivos de mi felici-dad.
Contents
Resumen iii
Abstract v
Agradecimientos vii
Contents ix
Resumen de la Introduccion xi
Introduction xix
Resumen del Capıtulo 1 1
1 Preliminaries 3
1.1 Bases and frames in Hilbert spaces . . . . . . . . . . . . . . . . . . . . . 3
1.2 Gramian operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Shift invariant spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Fiber space for shift invariant spaces . . . . . . . . . . . . . . . . . . . . 9
1.5 Riesz bases and frames for shift invariant spaces . . . . . . . . . . . . . . 11
Resumen del Capıtulo 2 15
2 Linear combinations of generators in systems of translates 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Minimal generator sets for finitely generated SIS in L2(Rd) . . . . . . . . 20
2.4 Eigenvalues of a conjugated matrix . . . . . . . . . . . . . . . . . . . . . 22
x CONTENTS
2.5 Linear combinations of frame generators in systems of translates in L2(Rd) 24
Resumen del Capıtulo 3 33
3 Extra invariance of shift invariant spaces 353.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Characterization of extra invariance of SIS in L2(Rd) . . . . . . . . . . . 39
Resumen del Capıtulo 4 43
4 Subspaces with extra invariance nearest to observed data 474.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3.1 Optimality for the class of SIS with extra-invariance . . . . . . . 55
4.3.2 Approximation with Paley-Wiener spaces . . . . . . . . . . . . . 58
4.3.3 The discrete case . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4 Approximations in the context of LCA groups . . . . . . . . . . . . . . . 67
Bibliography 69
Index 73
Introduccion(Resumen)
Los espacios invariantes por traslaciones enteras (EITE) son subespacios cerrados deL2(Rd) que resultan invariantes bajo traslaciones por numeros enteros. Estos espacios jue-gan un rol importante en la teorıa de aproximacion, analisis armonico, teorıa de wavelets,muestreo y procesamiento de imagenes [AG01, Gro01, HW96, Mal89]. Ademas, estosespacios sirven como modelos en varias aplicaciones en el procesamiento de senales eimagenes. Debido a su importancia tanto en la teorıa como en las aplicaciones, la estruc-tura de dichos espacios fue analizada en profundidad durante los ultimos veinticinco anos(ver por ejemplo [dBDVR94, dBDR94, Bow00, Hel64, RS95] y las referencias que enellos se encuentran).
Cualquier espacio invariante por traslaciones enteras puede generarse por un conjuntode funciones Φ en L2(Rd), en el sentido que el espacio coincide con la clausura del espaciogenerado por las traslaciones enteras de las funciones de Φ. En este caso, se dice que Φ
es un conjunto de generadores de V. Cuando existe un conjunto finito Φ de generadoresde V , decimos que V esta finitamente generado. En ese caso, existe un numero enteropositivo `, llamado la longitud de V , que se define como el mınimo numero de funcionesque generan V. Si Φ es un conjunto de generadores de V con ` elementos, decimos que Φ
es un conjunto minimal de generadores.
Un problema interesante en el contexto de los espacios invariantes por traslaciones en-teras finitamente generados es el siguiente: Supongamos que Φ = φ1, · · · , φm es unconjunto de generadores del espacio invariante por traslaciones V, ¿cuando es posibleobtener un conjunto minimal de generadores del conjunto de generadores dado Φ? Exis-ten varios ejemplos para los cuales ningun subconjunto de Φ es un conjunto minimal degeneradores. Luego, suprimir elementos de un conjunto de generadores dado Φ no pareceser un procedimiento exitoso.
Para dar solucion a este problema, Bownik y Kaiblinger en [BK06], mostraron quepuede obtenerse un conjunto de generadores minimal de V usando combinaciones linealesde los elementos del conjunto de generadores dado Φ. Mas aun, los autores probaron quecasi todo conjunto de ` funciones que son combinaciones lineales de φ1, · · · , φm es unconjunto de generadores minimal de V (ver [BK06, Theorem 1]). Es importante recalcar
xii Resumen de la Introduccion
aquı que por combinaciones lineales se entienden solo las que involucran a las funcionesφ1, · · · , φm y no a sus traslaciones.
Como las combinaciones lineales de un numero finito de funciones preservanpropiedades tales como la suavidad, el decaimiento, tener soporte compacto, ser de bandalimitada, etc., una consecuencia interesante del resultado de Bownik y Kaiblinger es quesi un conjunto de generadores de V tiene alguna propiedad adicional, existe un conjuntominimal de generadores del espacio que hereda dicha propiedad.
En muchos problemas que involucran a los espacios invariantes por traslaciones en-teras, es importante que el sistema de traslaciones Tkφ j : k ∈ Zd, j = 1, · · · ,m tengauna determinada estructura funcional como ser una base de Riesz, una base ortonormal oun marco. Luego, resulta interesante saber cuando puede obtenerse un conjunto minimalde generadores por combinaciones lineales de uno dado que preserve la misma estruc-tura. Mas precisamente, supongamos que Φ = φ1, · · · , φm genera el espacio invariantepor traslaciones enteras V de longitud `, y asumamos ademas que el nuevo conjunto degeneradores Ψ = ψ1, · · · , ψ` de V es producido tomando combinaciones lineales de lasfunciones de Φ. Es decir, supongamos que ψi =
∑mj=1 ai jφ j para i = 1, · · · , ` y para
algunos escalares complejos ai j. Sea A = ai ji, j ∈ C`×m la matriz formada por esos es-
calares, luego en notacion matricial, podemos escribir Ψ = AΦ. Queremos saber quematrices A preservan la estructura de Φ en Ψ. Mas precisamente, estudiamos el siguienteproblema: Supongamos que Tkφi : k ∈ Zd, i = 1, · · · ,m es un marco de V. ¿ Cuandosera Tkψi : k ∈ Zd, i = 1, · · · , ` tambien un marco de V?
En Chapter 2, damos respuesta a esta pregunta. Como mencionamos anteriormente, lapropiedad de ser un “conjunto de generadores” de un EITE V generalmente se preservapor la accion de una matriz A ([BK06]). Esto no sigue siendo valido para el caso demarcos.
Construımos un ejemplo de un espacio invariante por traslaciones enteras V con unconjunto de generadores Φ tal que sus traslaciones enteras Tkφi : k ∈ Zd, i = 1, · · · ,mforman un marco de V y tal que ninguna matriz A, de tamano `×m con ` < m, transformaa Φ en un nuevo conjunto de generadores que forme un marco de V.
En esta tesis, damos condiciones sobre la matriz A para que preserve la propiedad demarco. Estas condiciones las establecemos usando la nocion de angulo de Friedrich entresubespacios en espacios de Hilbert. Hay una relacion geometrica particular que debecumplirse entre el nucleo de A y el espacio columna de GΦ(ω), para casi todo ω. Parala demostracion de estos resultados usamos otros acerca de los valores singulares de lacomposicion de operadores. Ademas, damos una condicion analıtica entre la matriz A yGΦ(ω), equivalente a la anterior. A pesar de que estamos interesados en el caso ` = `(V)(la longitud del EITE V en estudio), la mayorıa de nuestros resultados siguen valiendoaun cuando `(V) ≤ ` ≤ m.
Por otro lado, incluimos los casos particulares de bases de Riesz y bases ortonormales,que son ya conocidos.
Luego, estudiamos un problema de aproximacion usando espacios invariantes por
Resumen de la Introduccion xiii
traslaciones enteras finitamente generados.
Una suposicion clasica en la teorıa de muestreo es que las senales a ser muestreadaspertenezcan a cierto espacio invariante por traslaciones enteras. Por ejemplo, si el espa-cio esta generado por φ(x) = sinc(x), el espacio es el espacio de Paley-Wiener PW defunciones de banda limitada definido de la siguiente manera
PW =f ∈ L2(R) : sop( f ) ⊆ [−1/2, 1/2]
. (0.1)
La razon para considerar dicho espacio se debe a un resultado clasico de Whittaker[Whit64] y Shannon [Sha49] que establece que una funcion f ∈ PW puede reconstru-irse exactamente a partir de sus muestras f (k) : k ∈ Z mediante la siguiente formula
f (k) =∑k∈Z
f (k)sinc(x − k). (0.2)
Como la funcion sinc tiene soporte infinito y decaimiento lento, el espacio PW de fun-ciones de banda limitada no es muy aplicable en implementaciones numericas. Luego,otros espacios son considerados para modelar senales. Por ejemplo, espacios invariantespor traslaciones enteras generados por funciones con soporte compacto o con buena lo-calizacion en tiempo y frecuencia. Uno de los objetivos en el problema de muestreo eneste contexto es buscar condiciones sobre los generadores de un espacio invariante portraslaciones enteras con el objetivo de que cualquier funcion del espacio pueda reconstru-irse a partir de sus valores en una sucesion de puntos, como en el caso de las funciones debanda limitada. Varios autores [AG01, Sun05, Wal92, ZS99] han estudiado este problemaen el contexto de los espacios invariantes por traslaciones enteras.
Nosotros consideramos el problema general de muestreo de senales que pertenecen a unconjunto finito de funciones F = f1, . . . , fm ⊆ L2(Rd). Ciertas suposiciones acerca de losespacios invariantes por traslaciones enteras son utiles en las aplicaciones, sin embargo, laeleccion de un espacio invariante por traslaciones enteras finitamente generado particularno se deduce tipicamente del conjunto de senales en F . Por ejemplo, las hipotesis a prioripueden ser correctas pero el conjunto de datos puede verse perturbado por ruido.
Con el objetivo de modelar conjuntos de senales por espacios invariantes por trasla-ciones enteras, que puedan ser maleables computacionalmente, es natural buscar espaciosinvariantes por traslaciones enteras V de longitud ` (donde tipicamente ` es chico com-parado con m) mas cercanos a un conjunto de datos dado, en el sentido que V minimiza
m∑j=1
‖ f j − PW f j‖2, (0.3)
sobre todos los espacios invariantes por traslaciones enteras W que pueden generarse pora lo sumo ` generadores. Aquı denotamos con PW a la proyeccion ortogonal sobre W.
En [ACHM07], los autores garantizan la existencia de dichos espacios invariantes portraslaciones enteras minimizantes. Dichos problemas de cuadrados mınimos pueden re-solverse usando la descomposicion en valores singulares (SVD). A pesar de que los es-pacios invariantes por traslaciones enteras son infinito dimensionales, y que la SVD no
xiv Resumen de la Introduccion
puede aplicarse en forma directa, los autores reducen dicho problema de optimizacion aun conjunto no contable de problemas de cuadrados mınimos finito dimensionales. Paradar solucion a estos problemas, los autores consideran una adaptacion del teorema deEckart-Young [EY36, Sch07]. Luego, construyen los generadores para el espacio optimousando las soluciones de los problemas reducidos. Ademas dan una formula para el erroren dicha aproximacion, usando una matriz especial.
Por otro lado, en [ACM08], los autores consideran el caso de espacios multiples en elcaso finito dimensional, encontrando una union de subespacios de dimension pequena quemejor aproximan a un conjunto de datos dado en Rd. Ademas, encuentran un algoritmopara encontrar dicha union. Este algoritmo es mejorado en [AACM11], usando tecnicasde reduccion dimensional.
En [AT11], los autores trabajan en el contexto general de espacios de Hilbert, encon-trando condiciones necesarias y suficientes para la existencia de subespacios optimos perosin mostrar una manera de construir dichos espacios. En Chapter 4 repasamos estos re-sultados previos junto con otros, en forma mas detallada.
El espacio de Paley-Wiener definido en (0.1) es un ejemplo de espacio invariante portraslaciones enteras en R. El mismo tiene la propiedad de ser no solo invariante por trasla-ciones enteras, si no de serlo ademas bajo cualquier traslacion real. Un teorema de Wiener(ver [Hel64]) caracteriza completamente a estos espacios invariantes por traslaciones en-teras en L2(Rd) que son invariantes bajo cualquier traslacion real, como aquellos de laforma
f ∈ L2(Rd) : sop f ⊆ Ω,
donde Ω ⊆ Rd es medible.
En muchas aplicaciones se busca que los espacios invariantes por traslaciones enterastengan extra invariancia [CS03, Web00], es decir que los mismos sean ademas invariantespor traslaciones sobre un particular conjunto M ⊆ Rd. En [ACHKM10] los autores car-acterizan aquellos espacios invariantes por traslaciones enteras con extra invariancia enL2(R). y en [ACP11] los autores generalizan estos resultados, dando una caracterizacionde estos espacios en L2(Rd). En Chapter 3 repasamos dichas caracterizaciones.
A pesar de que los autores en [ACHM07] muestran la existencia de un espacio invari-ante por traslaciones enteras finitamente generado V mas cercano al conjunto de datosF ⊆ L2(Rd), en el sentido de que V minimiza (0.3), los autores no dan respuesta si se lepide a los espacios invariantes por traslaciones enteras que tengan algun tipo de extra in-variancia. Luego, en esta tesis consideraremos este problema en un contexto mas general,es decir, espacios invariantes por traslaciones enteras con extra invariancia M, donde Mes un subgrupo cerrado de Rd que contiene exactamente a Zd.
En Chapter 4, primero consideramos el caso de M siendo un subgrupo propio de Rd
conteniendo a Zd. En este caso probamos que para cualquier conjunto de datos finitoF ⊆ L2(Rd), para cualquier subgrupo propio de Rd que contenga a Zd y para cualquier` ∈ N, existe un EITE V de longitud a lo sumo ` con extra invariancia M y tal queminimiza (0.3) sobre todos los espacios invariantes por traslaciones enteras de longitud
Resumen de la Introduccion xv
menor o igual que ` que son M extra-invariantes. Mas aun, construımos un conjunto degeneradores para el espacio optimo y damos una expresion para el error cometido en dichaaproximacion.
Luego consideramos el caso M = Rd, es decir, el caso de espacios invariantes portraslaciones enteras con extra invariancia total. Estos son espacios generados por lastraslaciones enteras de un numero finito de funciones que resultan Rd−invariantes. Eneste caso, como mencionamos antes, usando un teorema de Wiener [Hel64] sabemosque el espacio es isometrico a L2(Ω) para algun conjunto medible Ω ⊆ Rd. Estos sonespacios de Paley-Wiener generalizados. Con el fin de caracterizar la clase de subespaciosdonde trabajaremos, probamos que, para los espacios de Paley-Wiener generalizados, silas traslaciones enteras de los generadores forman una base de Riesz, entonces el conjuntoΩ asociado es un conjunto ` “multi-tile” de Rd, es decir∑
k∈Zd
χΩ(ω − k) = `, para casi todo ω ∈ Rd.
Estudiamos el problema de aproximacion para estos espacios de Paley-Wiener general-izados. Mostramos como construir un conjunto de generadores para el espacio optimoque ademas da una base de Riesz de traslaciones en Zd. Ademas mostramos una conexioncon resultados recientes acerca de la construccion de bases de exponenciales de L2(Ω)cuando Ω ⊆ Rd es un multi-tile (ver [GL14, Ko15]).
Finalmente consideramos un problema similar pero en el espacio de Hilbert `2(Zd). Laclase de espacios aproximantes en este caso es elegido convenientemente. Probamos laexistencia de un espacio optimo y mostramos como construir un conjunto de generadorespara dicho espacio.
El caso discreto resulta interesante en si mismo y es ademas relevante dada su conexioncon el caso continuo en L2(Rd), como veremos en el final de Chapter 4.
Esquema de la tesis
En Chapter 1 incluimos algunas definiciones basicas y resultados acerca de marcos ybases de Riesz en espacios de Hilbert que usaremos a lo largo de la tesis. Damos ademasciertas propiedades y caracterizaciones de los espacios invariantes por traslaciones en-teras. Definimos tambien el operador Gramiano, la funcion rango y la nocion de fibras enespacios invariantes por traslaciones enteras.
En Chapter 2 estudiamos condiciones para que un conjunto de generadores minimal deun espacio invariante por traslaciones enteras finitamente generado obtenido por combi-naciones lineales de uno dado preserve la misma estructura que el original.
Como mencionamos antes, Bownik y Kaiblinger en [BK06] mostraron que, dado unconjunto de generadores Φ de un EITE V, puede obtenerse un conjunto minimal de gen-eradores para V. Ademas, los autores probaron que casi todo conjunto de ` funciones
xvi Resumen de la Introduccion
que son combinaciones lineales de los elementos de Φ resulta un conjunto minimal degeneradores de V.
En varios problemas que involucran espacios invariantes por traslaciones enteras esimportante que el sistema de traslaciones enteras de los elementos del conjunto de gen-eradores Φ tenga una determinada estructura funcional (base de Riesz, base ortonormal,marco, etc). Luego, es interesante saber cuales de estas propiedades se preservan al tomarcombinaciones lineales. Es decir, supongamos que Φ es un conjunto de generadores de Vy que el nuevo conjunto de generadores Ψ es obtenido tomando combinaciones lineales delas funciones de Φ, es decir, Ψ = AΦ en notacion matricial, para una matriz A adecuada.En este captitulo damos condiciones para que la propiedad de marco sea preservada poruna matriz A. Ademas tratamos el mismo problema para el caso de bases de Riesz y basesortonormales.
En Chapter 3 repasamos algunas definiciones acerca de los espacios invariantes portraslaciones enteras con extra invariancia. Incluimos ademas resultados sobre la caracter-izacion de dichos espacios en L2(Rd).
Finalmente, en Chapter 4, dado un conjunto arbitrario de datos F = f1, . . . , fm ⊆
L2(Rd) probamos la existencia y como construir un espacio de dimension pequena mascercano a F . El espacio a construir debe ser invariante por traslaciones bajo un subgrupoaditivo fijado de Rd que contenga a Zd. Con pequena nos referimos a que nuestro espa-cio solucion debe estar generado por las traslaciones enteras de un pequeno numero degeneradores. Ademas, damos una expresion del error en terminos del conjunto de datosdado.
Por otro lado consideramos el problema de aproximar F por espacios generalizados dePaley-Wiener en Rd. En este caso, la solucion resulta invariante bajo cualquier traslacion.Mostramos la relacion entre estos espacios y los conjuntos “multi-tile” de Rd, y las conex-iones con resultados recientes acerca de la construccion de bases de Riesz de exponen-ciales.
Ademas consideramos un problema similar pero en el caso discreto de `2(Zd) y sobreuna cierta clase de subespacios apropiada.
Nuestros resultados pueden formularse en el contexto de grupos localmente compactosy abelianos. Este trabajo esta en proceso y no es parte de esta tesis.
Contribuciones originales y publicaciones de la tesis
Las contribuciones originales de la tesis se encuentran en Chapter 2 y en Chapter 4. EnChapter 2 presentamos los resultados acerca de la preservacion de la propiedad de marcopor combinaciones line ales, ver Section 2.5. En Chapter 4, mostramos los resultadosacerca de las aproximaciones con espacios invariantes por traslaciones enteros con extrainvariancia, Section 4.3. Estos nuevos resultados han originado las siguientes publica-ciones:
Resumen de la Introduccion xvii
• C. Cabrelli, C. Mosquera and V. Paternostro, Linear combinations of frame gener-ators in systems of translates, J. Math. Anal. Appl., 413(2), 2014, 776–788.
• C. Cabrelli and C. Mosquera, Subspaces with extra invariance nearest to observeddata, Preprint, 2015, http://arxiv.org/abs/1501.03187
Introduction
Shift invariant spaces (SISs) are closed subspaces of L2(Rd) that are invariant under integertranslations. They play an important role in approximation theory, harmonic analysis,numerical analysis, wavelet theory [AG01, Gro01, HW96, Mal89]. They also serve asmodels in many applications in signal and image processing.
Due to their importance in theory and applications, their structure has been deeplyanalyzed during the last twenty five years (see for example [dBDVR94, dBDR94, Bow00,Hel64, RS95] and the references therein).
Every shift invariant space V can be generated by a set Φ of functions in L2(Rd) in thesense that it is the closure of the space spanned by all integer translations of the functionsin Φ. In this case, the set Φ is called a set of generators for V. When there exists a finiteset of generators Φ for V , we say that V is finitely generated. In this case, there existsa positive integer `, called the length of V , that is defined as the minimal number offunctions that generate V. If the length of V is `, any set of generators with ` elements willbe called a minimal set of generators.
Concerning finitely generated shift invariant spaces, a particular problem of interest isthe following: suppose Φ = φ1, . . . , φm is a set of generators for a SIS V. When it ispossible to obtain a minimal set of generators from the given generators in Φ? There aremany examples with the property that no subset of Φ is a minimal set of generators. So,deleting elements from Φ may not be a successfull procedure.
To give solution to the above problem, Bownik and Kaiblinger in [BK06], showedthat a minimal set of generators for V can be obtained from Φ by linear combinations ofits elements. Moreover, they proved that almost every set of ` functions that are linearcombinations of φ1, . . . , φm is a minimal set of generators for V (see [BK06, Theorem1]). We want to remark here that the linear combinations of the generators only involvethe functions φ1, . . . , φm and not their translations.
As a consequence of the above Bownik and Kaiblinger’s result we have the followinginteresting observation. Suppose that a shift invariant space V has as a set of generatorsφ1, . . . , φm,which have some additional special properties, such as smoothness, compactsupport, bandlimitedness, decay, or membership in certain function spaces. Assume thatwe have that V can be generated by a fewer generators, that is, the length of V is ` < m.Then a priori such ` new generators may not have the special properties of the originalgenerators. Now by this result of Bownik and Kaiblinger we know that the answer is
xx Introduction
always affirmative. Moreover, we have that if the generators for V have some additionalproperty then there exists a minimal set of generators that inherits this property.
In many problems involving shift invariant spaces, it is important that the system oftranslates Tkφ j : k ∈ Zd, j = 1, . . . ,m bears a particular functional analytic structuresuch as being an orthonormal basis, a Riesz basis or a frame. Therefore, it is interestingto know when a minimal set of generators obtained by taking linear combinations ofthe original one has the same structure. More precisely, suppose that Φ = φ1, . . . , φm
generates a shift invariant space V of length `, and assume that a new set of generatorsΨ = ψ1, . . . , ψ` for V is produced by taking linear combinations of the functions in Φ.That is, assume that ψi =
∑mj=1 ai jφ j for i = 1, . . . , ` for some complex scalars ai j. If
we collect the coefficients in a matrix A = ai ji, j ∈ C`×m, then we can write in matrix
notation Ψ = AΦ. We would like to know, which matrices A transfer the structure ofΦ over Ψ. Precisely, in this thesis we study the following question: If we know thatTkφi : k ∈ Zd, i = 1, . . . ,m is a frame for V , when will Tkψi : k ∈ Zd, i = 1, . . . , ` alsobe a frame for V?
In Chapter 2, we answer this question completely. As we mentioned before, the prop-erty of being a “set of generators” for a SIS V is generically preserved by the action of amatrix A ([BK06]). This is not anymore valid for the case of frames. More than that, wewere able to construct a surprising example of a shift invariant space V of length ` with aset of generators Φ such that their integer translates Tkφi : k ∈ Zd, i = 1, . . . ,m form aframe for V and with the property that no matrix A, of size `×m with ` < m, transform Φ
into a new set of generators that form a frame for V .
We give exact conditions in order that the frame property is preserved by a matrix A.These conditions are established using the notion of Friedrichs angle between subspacesof a Hilbert space. The angle between the nullspace of A and the column space of thespecial matrix GΦ(ω) (the Gramian matrix associated to Φ) has to satisfy a particulargeometrical relation for almost all ω. The proof of these conditions use recent resultsabout singular values of the composition of operators. Also, we provide an equivalentanalytic condition between A and GΦ(ω) in order that this same result holds.
For completeness we also include the particular case of Riesz bases and orthonormalbases that are known.
Then we studied an approximation problem using finitely generated shift invariantspaces. In many signal and image processing applications, images and signals are as-sumed to belong to some shift invariant space. For example, if the space is generated byφ(x) = sinc(x), the space is the Paley-Wiener space PW of band-limited functions definedas follows
PW =f ∈ L2(R) : supp( f ) ⊆ [−1/2, 1/2]
. (0.4)
The reason for consider this space is a classical result of Whittaker [Whit64] and Shannon[Sha49] which states that a function f ∈ PW can be recovered exactly from its samples
Introduction xxi
f (k) : k ∈ Z by the following interpolation formula
f (k) =∑k∈Z
f (k) sinc(x − k). (0.5)
This result is fundamental in engineering and digital signal processing because it givesa framework for converting analog signals into sequences of numbers. These sequencescan then be processed digitally and converted back to analog signals via (0.5).
Since the sinc function has infinite support and slow decay, the space PW of band-limited functions is often unsuitable for numerical implementations. Then, other spacesare considered as signal models. For example, shift invariant spaces generated by func-tions with compact support or with good time-frequency localization. One of the goalsin the sampling problem in this context is finding conditions on the generators of a shiftinvariant space in order that every function of this space can be reconstructed from itsvalues in a discrete sequence of samples as in the band-limited case. Several authors[AG01, Sun05, Wal92, ZS99] have studied the sampling problem in the context of shiftinvariant spaces.
We consider the general problem of sampling signals that belong to a finite setF = f1, . . . , fm ⊆ L2(Rd). Although certain assumptions over shift invariant spaces areuseful in applications, the choice of the particular finitely generated shift invariant spacetypically is not deduced from the set of signals in F . For example, the a priori hypothesiscan be correct but the data can be corrupted by noise. Also, for computational considera-tions, a shift invariant space of length ` could be modelled by a shift invariant space withlength n much smaller than `. For example, in Learning Theory, the problem of reduc-ing the number of generators for a subspace of a reproducing kernel Hilbert space is alsoimportant for improving the efficiency and sparsity of learning algorithms (see [SZ04]).
In order to model sets of signals by finitely generated SIS in realistic cases, or to modelthese sets by a computationally tractable SIS, it is natural to search for a finitely generatedshift invariant space V with length at most ` (where typically ` is chosen to be smallcompared to m) that is nearest to a set of some observed data, in the sense that V minimizes
m∑j=1
‖ f j − PW f j‖2, (0.6)
over all shift invariant spaces W that can be generated by ` or less generators. Here PW
denotes the orthogonal projection over W.
In [ACHM07], the authors guarantee the existence of such shift invariant space mini-mizer. Least square problems of the form above in finite dimensional spaces can be solvedusing the singular value decomposition (SVD). Although shift invariant spaces are infi-nite dimensional and SVD cannot be applied directly, the authors reduce the optimizationproblem into an uncountable set of finite dimensional least square problems, using thespecial structure of shift invariant spaces. To give solution for these problems, the authorsconsider an adaptation of the Eckart-Young’s theorem [EY36, Sch07]. Then they con-struct the generators for an optimal shift invariant space for the original problem patching
xxii Introduction
together the solutions of the reduced problems. Also, the authors give a formula for theerror in this approximation using the eigenvalues of a special matrix that involves thefunctions in the set of data F .
On the other hand, in [ACM08] the authors consider the case of multiple subspaces inthe finitely dimensional case, finding a union of low dimensional subspaces that best fitsa given set of data in Rd. Besides they provide an algorithm to find it. This algorithm isimproved in [AACM11], using dimensional reduction techniques. Further, in [AT11], theauthors work in the general context of Hilbert spaces finding necessary and sufficient con-ditions for the existence of optimal subspaces but they do not provide a way to constructthem. We review these result joint with another related works in a more detailed way inChapter 4.
The Paley-Wiener space defined in (0.4) is an example of a shift invariant space in R. Ithas the property that is not only invariant under integer translations but it is invariant underevery real translation, that is, is a translation invariant SIS. A theorem of Wiener (see[Hel64]) completely characterizes the closed translation invariant subspaces of L2(Rd) asbeing of the form
f ∈ L2(Rd) : supp f ⊆ Ω,
where Ω ⊆ Rd is measurable.
In several applications it is desirable to have shift invariant spaces that possess extra-invariance [CS03, Web00], that is SIS that are not only invariant under integer translatesbut are also invariant under some particular set of translations M ⊆ Rd. In [ACHKM10]the authors characterize those invariant spaces with extra-invariance in L2(R) and in[ACP11] the authors generalizes these results giving a characterization of the extra-invariance for shift invariant spaces in L2(Rd). Further, in [ACP10] the authors give asimilar characterization for the context of shift invariant spaces on locally compact abelian(LCA) groups.
Although the authors in [ACHM07] show the existence of a finitely generated shiftinvariant space V nearest to a set of data F ⊆ L2(Rd), in the sense that V minimizes theexpression (0.6), it does not provide an answer if we require the shift invariant space tohave some type of extra-invariance. For this reason we consider this problem in this moregeneral context, that is, SIS with M extra-invariance, where M is a closed subgroup of Rd
that strictly contains Zd.
In Chapter 4, we first consider the case when M is a proper subgroup of Rd that containsZd. In this case we prove that for any finite set of dataF ⊆ L2(Rd), for any proper subgroupof Rd containing Zd and for any ` ∈ N, there exists a shift invariant space V of length atmost ` with extra-invariance M that minimizes (0.6) over all the SIS of length smaller orequal than ` that are M extra-invariant. Moreover, we construct a set of generators forthe SIS that produces the solution and give an expression for the exact value of the errorbetween the data F and the optimal subspace, using the eigenvalues of a special matrix.
It is worth to mention that a previous result in this direction appears in [AKTW12]where the authors consider the problem for principal shift invariant spaces in one variable
Introduction xxiii
and assuming that the space has an orthonormal generator (that is, a generator with or-thonormal integer translates). These are keys for their proofs. So the techniques of thisparticular case do not apply to our general case.
Next we consider the case of M = Rd, that is, the case of shift invariant spaces withtotal extra-invariance. They are spaces generated by integer translates of a finite number offunctions that result Rd−invariant. In this case, as we mentioned before, using a theoremof Wiener [Hel64] we know that the space is isometric to L2(Ω) for some measurablesubset Ω ⊆ Rd. These are generalized Paley-Wiener spaces. In order to characterize theclass where we will consider the approximation problem, we prove for the generalizedPaley-Wiener spaces that, if the integer translates of the generators form a Riesz basisthen the associated set Ω is an ` multi-tile of Rd, that is,∑
k∈Zd
χΩ(ω − k) = `, for a.e. ω ∈ Rd.
We study the approximation problem for these generalized Paley-Wiener spaces. Weshow how to construct a set of generators for the optimal SIS that also gives a Rieszbasis of translates in Zd. Also, we show a connection with recent results concerning theconstruction of bases of exponentials for L2(Ω) when Ω ⊆ Rd is a multi-tile set (see[GL14, Ko15]).
Finally, we consider a similar problem but in the Hilbert space `2(Zd). The class of ap-proximating subspaces will be chosen conveniently. We prove the existence of an optimalsubspace and show how to construct a set of generators for this space, similarly to theresults obtained for L2(Rd).
Although this discrete case is interesting by itself, it is also relevant due to its connec-tion with the continuous case in L2(Rd), as we will show at the end of Chapter 4.
Thesis outline
In Chapter 1 we include basic definitions and results regarding frames and Riesz basesin Hilbert spaces that we will use throughout this thesis. We give some characterizationsand properties of shift invariant spaces. We also define the Gramian operator, the rangefunction and the notion of fibers for shift invariant spaces.
In Chapter 2 we study conditions in order that a minimal set of generators of a finitelygenerated shift invariant space obtained by taking linear combinations of the original onehas the same structure.
As we mentioned above, Bownik and Kaiblinger in [BK06] showed that a minimalset of generators for a shift invariant space V can be obtained from the given set of gen-erators Φ. Furthermore, they proved that almost every set of ` functions that are linearcombinations of the elements of Φ is a minimal set of generator for V.
xxiv Introduction
In several problems involving shift invariant spaces it is important that the system ofinteger translates of the elements of a generator set Φ bears a particular structure suchas being an orthonormal basis, a Riesz basis or a frame. Then it is interesting to analizewhen these functional analytic structures are preserved by taking linear combinations ofthe given set of generators. More precisely, assume that Φ is a set of generators for Vand suppose that a new set of generators Ψ is obtained by taking linear combinationsof the functions in Φ, that is, in matrix notation, Ψ = AΦ for an appropriate matrix A.Then we give exact conditions in order that the frame property is preserved by a matrix A.Surprisingly the results are very different to the recently studied case when the propertyto be a frame is not required. Also, we include the case of Riesz bases and orthonormalbases that are known.
In Chapter 3 we give definitions and properties concerning shift invariant spaces withextra invariance. We review results about the characterization of shift invariant spacesthat are not only invariant under integer translations, but are also invariant under a givenclosed subgroup of Rd.
In Chapter 4, given an arbitrary finite set of dataF = f1, . . . , fm ⊆ L2(Rd) we prove theexistence and show how to construct a “small shift invariant space” that is the “closest” tothe data F over certain class of closed subspaces of L2(Rd). The approximating subspaceis required to have extra-invariance properties, that is to be invariant under translationsby a prefixed additive subgroup of Rd containing Zd. This is important for example insituations where we need to deal with jitter error of the data. Here small means thatour solution subspace should be generated by the integer translates of small number ofgenerators. We give an expression for the error in terms of the data and construct aParseval frame for the optimal space.
We also consider the problem of approximating F from generalized Paley-Wienerspaces of Rd, that are generated by the integer translates of a finite number of functions.These spaces can be seen as finitely generated shift invariant spaces that are Rd invariant.
We show the relations between theses spaces and multi-tile sets of Rd, and the connec-tions with recent results on Riesz basis of exponentials.
Further we consider the same approximation problem in a discrete case, that is, in theHilbert space `2(Zd) and over a certain appropriated class of subspaces.
Original contributions and publications from this thesis
The original contributions of this thesis are in Chapter 2 and Chapter 4. In Chapter 2 wepresent the results about the preservation of the frame property by linear combinations,see Section 2.5. In Chapter 4, we have the results about approximations by shift invariantspaces with extra invariance, Section 4.3. These new results have originated the followingpublications:
Introduction xxv
• C. Cabrelli, C. Mosquera and V. Paternostro, Linear combinations of frame gener-ators in systems of translates, J. Math. Anal. Appl., 413(2), 2014, 776–788.
• C. Cabrelli and C. Mosquera, Subspaces with extra invariance nearest to observeddata, Preprint, 2015, http://arxiv.org/abs/1501.03187
Capıtulo 1: Preliminares(Resumen)
En este capıtulo presentamos las notaciones y definiciones que usaremos a lo largo de estatesis. Algunos de los resultados que presentaremos son ya conocidos.
En la Seccion 1.1, comenzamos dando definiciones y algunas propiedades importantesacerca de bases ortonormales, bases de Riesz y marcos en espacios de Hilbert.
En la Seccion 1.2, presentamos la nocion de operador Gramiano.
En la Seccion 1.3 revisamos definiciones y algunas propiedades acerca de los espaciosinvariantes por traslaciones enteras.
En la Seccion 1.4 presentamos las nociones de funcion rango, fibras y espacio fibradoen el contexto de espacios invariantes por traslaciones enteras.
En la Seccion 1.5 caracterizamos cuando un sistema de traslaciones enteras de un con-junto Φ ⊆ L2(Rd) es una familia de Bessel, un marco o una base de Riesz, en termino delas fibras y del operador Gramiano.
1
Preliminaries
In this chapter we introduce the notation and some definitions used throughout this thesis.Some of the results presented below are well known, but we include them here for thesake of completeness.
In Section 1.1, we begin by giving definitions and some relevant properties about or-thonormal bases, Riesz bases and frames in Hilbert spaces. In Section 1.2 we introducethe notion of Gramian operator. In Section 1.3 we present the definition and some prop-erties of shift invariant spaces. In Section 1.4 we give the definitions of range function,fibers and fiber spaces in the context of shift invariant spaces. In Section 1.5 we character-ize the system of integer translates of a set Φ ⊆ L2(Rd) as being a Bessel family, a frame,or a Riesz basis in terms of fibers and in terms of its Gramian operator.
1.1 Bases and frames in Hilbert spaces
In this section we collect some definitions about bases and frames in Hilbert spaces. Formore details we refer the reader to [Chr03, Hei11] and the references therein.
Definition 1.1.1. Let H be a separable Hilbert space. A sequence of vectors f j j∈J be-longing to H is a (Schauder) basis for H if, for each f ∈ H, there exist unique scalarcoefficients c j( f ) j∈J such that
f =∑j∈J
c j( f ) f j. (1.1)
Sometimes we refer to (1.1) as the expansion of f in the basis f j j∈J and this equationmerely means that the series f =
∑j∈J c j( f ) f j converges with respect to the chosen order
of the elements.
Besides the existence of an expansion of each f ∈ H, the above definition asks foruniqueness. For example, if the basis f j j∈J is orthonormal (that is, 〈 fi, f j〉 = δi, j for alli, j ∈ J), there is a unique representation of each f ∈ H in the basis, being c j( f ) = 〈 f , f j〉
for all j ∈ J.
4 Preliminaries
Theorem 1.1.2. Let H be a separable Hilbert space and f j j∈J be an orthonormal systemin H. The following are equivalent:
(i) f j j∈J is an orthonormal basis.
(ii) f =∑
j∈J〈 f , f j〉 f j, for all f ∈ H.
(iii)∑
j∈J |〈 f , f j〉|2 = ‖ f ‖2, for all f ∈ H (Parseval’s identity).
(iv) span f j j∈J = H.
(v) If 〈 f , f j〉 = 0, for all j ∈ J, then f = 0.
Theorem 1.1.3. Every separable Hilbert space H has an orthonormal basis.
The simplest case is the following.
Example 1.1.4. Let f j be the sequence in `2(J) whose j−th entry is 1, and all other entriesare zero. Then f j j∈J is an orthonormal basis for `2(J) and it is called the canonicalorthonormal basis.
Now we will introduce the definition of Riesz bases.
Definition 1.1.5. Let H be a separable Hilbert space and f j j∈J be a sequence in H. Thesequence f j j∈J is said to be a Riesz basis for H if it is complete in H and if there exist0 < α ≤ β such that
α∑j∈J
|c j|2 ≤ ‖
∑j∈J
c j f j‖2 ≤ β
∑j∈J
|c j|2, ∀ c j j∈J ∈ `
2(J). (1.2)
The constants α and β are called Riesz bounds.
A sequence f j j∈J satisfying (1.2) for all sequences c j j∈J is called a Riesz sequence.Then, a Riesz sequence f j j∈J is a Riesz basis for span f j j∈J, which might just be asubspace of H.
The following theorem establishes a relationship between Riesz bases, bases and or-thonormal bases. As a consequence of this result we can obtain that all orthonormal basesare Riesz bases.
Theorem 1.1.6. Let f j j∈J be a sequence in H. The following statements are equivalent.
(i) f j j∈J is a Riesz basis for H.
(ii) f j j∈J is a basis for H, and∑j∈J
c j f j converges if and only if∑j∈J
|c j|2 < ∞.
1.1 Bases and frames in Hilbert spaces 5
(iii) There exist a bounded linear operator T : H → H and an orthonormal basis x j j∈J
of H such that T (x j) = f j for all j ∈ J.
The following proposition gives a condition for a Riesz basis to be an orthonormalbasis.
Proposition 1.1.7. Assume that span f j j∈J = H and that
‖∑j∈J
c j f j‖2 =
∑j∈J
|c j|2
for all finite scalar sequences c j j∈J. Then f j j∈J is an orthonormal basis for H.
As a generalization of the basis concept, we will give the definition of frame.
The main feature of a basis f j j∈J in a Hilbert space H is that every f ∈ H can berepresented as an (infinite) linear combination of the elements f j in the basis. The coeffi-cients c j( f ) in (1.1) are unique. We now introduce the concept of frames. A frame is alsoa sequence of elements f j j∈J in H, which allows every f ∈ H to be written as in (1.1).However, the corresponding coefficients are not necessarily unique. Although the lackof uniqueness, it is usually enough to know the existence of some usable coefficients, to-gether with a recipe for finding them. For example, in 1986, Daubechies, Grossmann andMeyer [DGM86] observed that frames can be used to find series expansions of functionsin L2(R) which are very similar to the expansions using orthonormal bases.
Definition 1.1.8. The sequence f j j∈J is said to be a frame for H if there exist 0 < α ≤ βsuch that
α‖ f ‖2 ≤∑j∈J
|〈 f , f j〉|2 ≤ β‖ f ‖2, ∀ f ∈ H. (1.3)
The constants α and β are called frame bounds. They are not unique. The optimal upperframe bound is the infimum over all upper frame bounds, and the optimal lower framebound is the supremum over all lower frame bounds.
When only the right hand side inequality in (1.3) is satisfied, we say that f j j∈J is aBessel sequence with Bessel bound β.
We say that a frame is tight if we can choose α = β as frame bounds, and it is a Parsevalframe if α = β = 1.
If a frame ceases to be a frame when an arbitrary element is removed, it is called anexact frame.
We say that f j j∈J is a frame sequence if it is a frame for span f j j∈J.
Example 1.1.9. Let e j j∈J be an orthonormal basis for a separable Hilbert space H.
(a) By Plancherel’s identity, e j j∈J is a tight frame with α = β = 1. Moreover, e j j∈J
is an exact frame since if we delete any element e j′ , then∑
j, j′ |〈e j, e j′〉|2 = 0, and
therefore e j j, j′ cannot be a frame.
6 Preliminaries
(b) e1, e1, e2, e2, . . . is a tight inexact frame with bounds α = β = 2, but it is not abasis, although it does contain an orthonormal basis. Similarly, if f j j∈J is anotherorthonormal basis for H then e j j∈J ∪ f j j∈J is a tight inexact frame.
(c) e1, e2/2, e3/3, . . . is a complete orthogonal sequence and it is a basis for H, but itdoes not possess a lower frame bound and hence is not a frame.
Usually, in this thesis we will work with the above definitions in the following context.We will assume that H is a closed subspace of L2(Rd) and the sequence fkk∈Z ⊆ Hconsists of integer translates of a fixed finite set of functions Φ ⊆ L2(Rd).
As we mention above, frames can be viewed as a generalization of Riesz bases.
Theorem 1.1.10. A Riesz basis f j j∈J for H is a frame for H, and the Riesz basis boundscoincide with the frames bounds.
A frame f j j∈J which is not a basis is said to be overcomplete, that is, there existcoefficients c j j∈J ∈ `
2(J)\ 0 for which∑
j∈J c j f j = 0. The next theorem gives necessaryand sufficient conditions in order for a frame to be a Riesz basis.
Theorem 1.1.11. Let f j j∈J be a sequence in H. Then , f j j∈J is a Riesz basis if and onlyif it is an exact frame for H.
Now we define some operators that we will use later.
Definition 1.1.12. Let f j j∈J be a Bessel sequence in a Hilbert space H.
(a) The analysis operator for f j j∈J is the continuous mapping U : H → `2(J) definedby
U f = (〈 f , f j〉) j∈J for f ∈ H.
(b) The synthesis operator or the pre-frame operator of f j j∈J is the continuous map-ping U∗ : `2(J)→ H defined by
U∗(c j j∈J) =∑j∈J
c j f j for c j j∈J ∈ `2(J).
(c) The frame operator for f j j∈J is the continuous mapping S : H → H defined by
S f = U∗U f =∑j∈J
〈 f , f j〉 f j, f ∈ H.
Since f j j∈J is a Bessel sequence, the synthesis operator is bounded from `2(J) to H.
Using the synthesis operator we can characterize frames as the following theoremshows.
Theorem 1.1.13. A Bessel sequence f j j∈J in H is a frame for H if and only if its synthesisoperator U∗ : `2(J)→ H is a well-defined surjective operator from `2(J) into H.
1.1 Bases and frames in Hilbert spaces 7
For an arbitrary sequence f j j∈J in a Hilbert space, we observe that span f j j∈J is itselfa Hilbert space, where span f j j∈J denotes the vector space consisting of all finite linearcombinations of vectors f j. Therefore, the above theorem leads to a statement about framesequences.
Corollary 1.1.14. A sequence f j j∈J in H is a frame sequence if and only if its synthesisoperator U∗ : `2(J)→ H is a well-defined operator on `2(J) and has closed range.
Remark 1.1.15. As a consequence of the above proposition we have that if f j j∈J is aframe for S = span f j j∈J, then
S =
∑j∈J
c j f j : c j j∈J ∈ `2(J)
.In the next theorem we will present properties about the frame operator that allow us
characterize a frame.
Theorem 1.1.16. Given a sequence f j j∈J in a Hilbert space H, the following statementsare equivalent.
(a) f j j∈J is a frame with bounds α and β.
(b) The frame operator S is bounded, invertible, self-adjoint, positive, and satisfies
αI ≤ S ≤ βI.
(c) S −1 f j j∈J is a frame for H, with frame bounds 0 < β−1 < α−1 and frame operatorS −1.
(d) The following series converges unconditionally for each f ∈ H :
f =∑j∈J
〈 f , S −1 f j〉 f j =∑j∈J
〈 f , f j〉S −1 f j. (1.4)
(e) If the frame is tight, then S = αI, S −1 = α−1I and
f = α−1∑j∈J
〈 f , f j〉 f j ∀ f ∈ H.
Given f j j∈J be a frame in H, a Bessel sequence g j j∈J is said to be a dual frame of f j j∈J if
f =∑j∈J
〈 f , g j〉 f j, ∀ f ∈ H.
The frame S −1 f j j∈J is a dual frame of f j j∈J, called the canonical dual frame of f j j∈J.If f j j∈J is a Riesz basis, the unique dual is S −1 f j j∈J. When the frame is overcomplete,
8 Preliminaries
that is, is not a Riesz basis, there exist dual frames which are different from the canonicaldual.
The frame decomposition is the most important frame result. That is a natural reasonto view a frame as a some kind of generalized basis. As (1.4) shows, if f j j∈J is a framefor H, then every element in H has a representation as an infinite linear combination ofthe frame elements. Then, all the information about a given f ∈ H is contained in thesequence 〈 f , S −1 f j〉 j∈J. The numbers 〈 f , S −1 f j〉 are called frame coefficients.
1.2 Gramian operator
In this section we introduce the notion of Gramian associated with a Besel sequence in aHilbert space H.
Definition 1.2.1. Let f j j∈J be a Bessel sequence in a Hilbert space H and let U : H →`2(J) be the analysis operator of f j j∈J. The Gramian of f j j∈J is defined by
G : `2(J)→ `2(J), G B UU∗. (1.5)
We can identify the Gramian with its matrix representation
(G)i, j = 〈 fi, f j〉, ∀i, j ∈ J.
Also, we can define the dual Gramian of f j j∈J by
G : H → H, G B U∗U.
We observe that one can consider the Gramian associated to any sequence f j j∈J in H,but if we want to define a bounded operator on `2(J) we can not avoid the Bessel conditionas the following lemma shows.
Lemma 1.2.2. For a sequence f j j∈J in H the following are equivalent:
(a) f j j∈J is a Bessel sequence with bound β.
(b) The Gramian associated to f j j∈J defines a bounded operator on `2(J), with normat most β.
Given a Hilbert space H and a bounded linear operator T : H → H, we will denote byΣ(T ) the spectrum of T, that is,
Σ(T ) = λ ∈ C : λI − T is not invertible,
where I denotes the identity operator of H.
We can relate the notion of frame sequence with the spectrum of its Gramian as follows.
Theorem 1.2.3. Let f j j∈J ⊆ H be a Bessel sequence and G be its Gramian operator.Then f j j∈J is a frame sequence with constants α and β if and only if
Σ(G) ⊆ 0 ∪ [α, β].
1.3 Shift invariant spaces 9
1.3 Shift invariant spaces
In this section, we will give some definitions and properties of shift invariant spaces. Formore details we refer to [Bow00, dBDR94, dBDVR94, Hel64, RS95] and the referencestherein.
Definition 1.3.1. A closed subspace V ⊆ L2(Rd) is shift invariant if f ∈ V implies thatTk f ∈ V for any k ∈ Zd. Here Ty f (x) = f (x − y) is the translation by the vector y ∈ Rd.
For any subset Φ ⊆ L2(Rd) we define
S (Φ) = spanTkφ : φ ∈ Φ, k ∈ Zd and E(Φ) = Tkφ : φ ∈ Φ, k ∈ Zd.
We call S (Φ) the shift invariant space (SIS) generated by Φ. We call Φ a set of gener-ators for V(Φ). If Φ = φ we will also write S (φ) = S (φ). If V = S (Φ) for some finiteset Φ we say that V is a finitely generated SIS, and a principal SIS if V can be generatedby the integer translates of a single function.
For a finitely generated shift invariant space V ⊆ L2(Rd) we define the length of V as
`(V) = minn ∈ N : ∃ φ1, . . . , φn ∈ V with V = S (φ1, . . . , φn).
We say that Φ is a minimal set of generators for V if V = S (Φ) and Φ has exactly `(V)elements.
A shift invariant space always has a Parseval frame of translates as the following theo-rem shows.
Theorem 1.3.2. Given V ⊆ L2(Rd) a SIS, there exists a subset Φ = φ j j∈J ⊆ V such thatE(Φ) is a Parseval frame for V. If V is finitely generated, the cardinal of J can be chosento be the length of V.
We should mention here that there are shift invariant spaces which have no Riesz basesof translates. That is one of the reasons for which it is important to consider frames insteadof Riesz bases for studying the structure of shift invariant spaces.
1.4 Fiber space for shift invariant spaces
Helson in [Hel64] introduced range functions and used this notion to completely char-acterize shift invariant spaces. Later on, several authors have used this frameworkto describe and characterize frames and bases of these spaces. See for example[dBDVR94, dBDVR94, RS95, Bow00, CP10]. We are not going to review the completetheory of Helson. We will only mention the required definitions and properties that weneed in this thesis. We refer to [Bow00] for a clear and complete description.
10 Preliminaries
Let U ⊆ Rd be a measurable set of representatives of the quotient Rd/Zd (we usu-ally will take U = [−1/2, 1/2]d or U = [0, 1]d). The Hilbert space of square inte-grable vector functions L2(U, `2(Zd)) consists of all vectors valued measurable functionsF : U → `2(Zd) with the norm
‖F‖ =
(∫U
‖F(x)‖2`2(Zd) dx
)1/2
< ∞.
Proposition 1.4.1. The function τ : L2(Rd)→ L2(U, `2(Zd)) defined for f ∈ L2(Rd) by
τ f : U → `2(Zd), τ f (ω) ≡ f (ω + k)k∈Zd ,
is an isometric isomorphism between L2(Rd) and L2(U, `2(Zd)).
The sequence τ f (ω) ≡ f (ω + k)k∈Zd is called the fiber of f at ω.
Definition 1.4.2. A range function is a mapping
J : U → closed subspaces of `2(Zd).
J is measurable if the associated orthogonal projections P(ω) : `2(Zd) → J(ω) areweakly operators measurable. Using that in a Hilbert space measurability is equivalentto weak measurability, the measurability of J es equivalent to ω 7→ P(ω)a being vectormeasurable for each a ∈ `2(Zd).
The next proposition gives a characterization of shift invariant spaces in terms of therange function.
Proposition 1.4.3. A closed subspace V ⊆ L2(Rd) is shift invariant if and only if
V = f ∈ L2(Rd) : τ f (ω) ∈ JV(ω) for a.e. ω ∈ U,
where JV is a measurable range function. The correspondence between V and JV is one-to-one under the convention that the range functions are identified if they are equal a.e.Furthermore, if V = S (Φ) for some countable Φ ⊆ L2(Rd), then
JV(ω) = spanτϕ(ω) : ϕ ∈ Φ, for a.e. ω ∈ U.
The subspace JV(ω) is called the fiber space of V at ω.
We observe that if V = S (ϕ1, . . . , ϕm) ⊆ L2(Rd) then
JV(ω) = span τϕ1(ω), . . . , τϕm(ω).
Then the fiber spaces JV(ω) are all finite dimensional subspaces of `2(Zd).
We have the following useful properties.
Lemma 1.4.4. If f ∈ L2(Rd), then
1.5 Riesz bases and frames for shift invariant spaces 11
(i) the sequence τ f (ω)k = f (ω+k)k is a well-defined sequence in `2(Zd) a.e. ω ∈ Rd.
(ii) ‖τ f (ω)‖`2 is a measurable function of ω and
‖ f ‖2 = ‖ f ‖2 =
∫U
‖τ f (ω)‖2`2 dω.
Lemma 1.4.5. Let V be a finitely generated SIS in L2(Rd). Then we have
(i) JV(ω) is a closed subspace of `2(Zd) for a.e. ω ∈ U.
(ii) V = f ∈ L2(Rd) : τ f (ω) ∈ JV(ω) for a.e. ω ∈ U.
(iii) For each f ∈ L2(Rd) we have that ‖τ(PV f )(ω)‖`2 is a measurable function of thevariable ω and
τ(PV f )(ω) = PJV (ω)(τ f (ω)).
(iv) Let ϕ1, . . . , ϕm ∈ L2(Rd). We have that ϕ1, . . . , ϕm is a set of generators of V, if andonly if τϕ1(ω), . . . , τϕm(ω) span JV(ω) for a.e. ω ∈ U.
Now we introduce the concept of dimension function of a shift invariant space.
Definition 1.4.6. The dimension function of V is a mapping given by
dimV : U → N ∪ 0,∞, dimV(ω) = dim JV(ω),
where JV is the range function associated with V.
We have the following property concerning the length of a finitely generated SIS.
Proposition 1.4.7. Let V ⊆ L2(Rd) be a finitely generated SIS. Then
length (V) = ess supdimV(ω) : ω ∈ U.
1.5 Riesz bases and frames for shift invariant spaces
Next results give a characterization of system of translates E(Φ) as being a Bessel se-quence, a frame, or a Riesz basis using the notion of fibers.
Theorem 1.5.1. Suppose Φ ⊆ L2(Rd) is countable.
(a) E(Φ) is a Bessel sequence in L2(Rd) with bound β if and only if τϕ(ω) : ϕ ∈ Φ is aBessel sequence in `2(Zd) with bound β for a.e. ω ∈ U.
(b) E(Φ) is a frame for V = S (Φ) with bounds α and β if and only if τϕ(ω) : ϕ ∈ Φ isa frame for JV(ω) with bounds α and β for a.e. ω ∈ U.
12 Preliminaries
(c) E(Φ) is a Riesz basis for V = S (Φ) with bounds α and β if and only if τϕ(ω) : ϕ ∈Φ is a Riesz basis for JV(ω) with bounds α and β for a.e. ω ∈ U.
We observe that the above theorem allows us to reduce the problem of checking whetherE(Φ) is a frame or a Riesz basis in a big subspace of L2(Rd) to analyzing the fibersin smaller subspaces of `2(Zd). Then, although the finitely generated SISs are infinitedimensional subspaces, some important properties can be translated into properties on thefibers of the spanning sets, working with finite dimensional subspaces of `2(Zd).
Now we define de Gramian operator for shift invariant spaces.
Definition 1.5.2. Let Φ = ϕ j j∈J be a countable collection of functions in L2(Rd) suchthat E(Φ) is a Bessel sequence. The Gramian GΦ of Φ at ω ∈ U is GΦ(ω) : `2(J)→ `2(J),
[GΦ(ω)]i j = 〈τϕ j(ω), τϕi(ω)〉`2(Zd) =∑k∈Zd
ϕi(ω + k) ϕ j(ω + k), ∀i, j ∈ J. (1.6)
The dual Gramian GΦ of Φ at ω ∈ U is GΦ(ω) : `2(Zd)→ `2(Zd),
[GΦ(ω)]kl =∑j∈J
ϕ j(ω + k) ϕ j(ω + l), ∀k, l ∈ Zd. (1.7)
We observe that the above definition agrees with the Gramian operator associated tothe Bessel sequence τΦ(ω) = τϕ j(ω) j∈J in `2(Zd), with the notation of Definition 1.5.
The Gramian of Φ is determined a.e. by its values at ω ∈ U and satisfies GΦ(ω)∗ =
GΦ(ω) for a.e. ω ∈ U.
Now, using the notion of Gramian operator, we obtain the following characterization offrames and Riesz bases. This result can be proved using Theorem 1.5.1. For more detailssee [Bow00].
Theorem 1.5.3. Let Φ = ϕ j j∈J ⊆ L2(Rd). Then:
(1) The following statements are equivalent:
(a) E(Φ) is a Bessel sequence with bound β.
(b) ess supω∈U ‖GΦ(ω)‖op ≤ β.
(2) The following statements are equivalent:
(a) E(Φ) is a frame for S (Φ) with bounds α and β.
(b) For almost all ω ∈ U,
α〈GΦ(ω)a, a〉 ≤ 〈G2Φ(ω)a, a〉 ≤ β〈GΦ(ω)a, a〉, ∀a ∈ `2(J).
(c) For almost all ω ∈ U, Σ(GΦ(ω)) ⊆ [α, β] ∪ 0.
1.5 Riesz bases and frames for shift invariant spaces 13
(3) The following statements are equivalent:
(a) E(Φ) is a Riesz basis for S (Φ) with bounds α and β.
(b) For almost all ω ∈ U,
α‖a‖2 ≤ 〈GΦ(ω)a, a〉 ≤ β‖a‖2, ∀a ∈ `2(J).
(c) For almost all ω ∈ U, Σ(GΦ(ω)) ⊆ [α, β].
(d) For almost all ω ∈ U, ‖GΦ(ω)‖op ≤ β and ‖(GΦ(ω))−1‖op ≤ 1α.
For a finitely generated SIS V , the length of V can be expressed in terms of the Gramianas follows (see [Bow00, dBDVR94, TW14])
`(V) = ess supω∈U
rk(GΦ(ω)) (1.8)
where rk(B) denotes the rank of a matrix B and Φ is a generator set for V .
Capıtulo 2:Combinaciones lineales de generadores en
sistemas de traslaciones(Resumen)
Sea V un espacio invariante por traslaciones enteras generado por las traslaciones enterasde Φ = φ1, . . . , φm. Es decir, V es la clausura del conjunto de todas las funciones ϕ de laforma
ϕ(x) =
m∑j=1
∑k∈Zd
c j,k φ j(x − k), x ∈ Rd, (1.9)
donde solo un numero finito de los coeficientes c j,k ∈ C son no nulos.
Resulta interesante saber cuando es posible obtener un conjunto minimal de gener-adores del conjunto de generadores dado Φ. Es decir, cuando existen ψ1, . . . , ψ` ⊆L2(Rd) con ` = long(V) ≤ m tales que
S (φ1, . . . , φm) = S (ψ1, . . . , ψ`).
Hay varios ejemplos en los cuales ningun subconjunto de Φ resulta un conjunto de gener-adores minimal. Luego, suprimir elementos de Φ no parece ser un procedimiento exitoso.
En busca de dar solucion a esta pregunta, Bownik y Kaiblinger en [BK06], mostraronque puede obtenerse de Φ un conjunto minimal de generadores para V usando combina-ciones lineales de sus elementos. Mas aun, los autores prueban que casi todo conjunto de` funciones que son combinaciones lineales de φ1, . . . , φm resulta un conjunto minimalde generadores de V (Ver Section 2.3). Es importante recalcar aquı que las combinacioneslineales solo involucran a las funciones φ1, . . . , φm y no a sus traslaciones.
En muchos problemas que involucran a los espacios invariantes por traslaciones en-teras es importante que el sistema de traslaciones Tkφ j : k ∈ Zd, j = 1, . . . ,m tengauna determinada estructura analıtica como ser una base ortonormal, una base de Rieszo un marco. Luego, resulta interesante saber cuando un conjunto minimal de gener-adores obtenido por combinaciones lineales de los elementos de uno dado preserva la
16 Resumen del Capıtulo 2
misma estructura. Mas precisamente, supongamos que Φ = φ1, . . . , φm genera un es-pacio invariante por traslaciones enteras V de longitud `, y supongamos que el nuevoconjunto de generadores Ψ = ψ1, . . . , ψ` para V se forma tomando combinaciones lin-eales de las funciones de Φ. Es decir, supongamos que ψi =
∑mj=1 ai jφ j para i = 1, . . . , ` y
para ciertos escalares complejos ai j. Si armamos con los coeficientes la siguiente matrizA = ai ji, j ∈ C
`×m, podemos escribir matricialmente Ψ = AΦ. Luego, queremos saberque matrices A preservan la estructura que tenıa Φ en Ψ. Concretamente, estudiamos lasiguiente pregunta: Supongamos que Tkφi : k ∈ Zd, i = 1, . . . ,m es un marco de V.¿Cuando sera Tkψi : k ∈ Zd, i = 1, . . . , ` tambien un marco de V?
En la Section 2.5 damos respuesta a la anterior pregunta.
Como mencionamos antes, la propiedad de ser un “conjunto de generadores” de unespacio invariante por traslaciones enteras V se preserva generalmente bajo la accion deuna matriz A ([BK06]). Esto no sigue sucediendo para el caso de marcos, lo cual esbastante inesperado. Mas aun, en este capıtulo construımos un sorprendente ejemplo deun espacio invariante por traslaciones enteras V con un conjunto de generadores Φ tal queel sistema formado por sus traslaciones enteras Tkφi : k ∈ Zd, i = 1, . . . ,m forma unmarco de V pero tal que ninguna matriz A, de tamano `×m con ` < m, transforma a Φ enun nuevo conjunto de generadores que forme un marco de V.
Nuestros principales resultados dan condiciones exactas sobre la matriz A para quedicha matriz preserve la propiedad de ser un marco del espacio. Estas condiciones estanexpresadas en funcion de una relacion geometrica particular que debe cumplirse entre elnucleo de A y el espacio columna de GΦ(ω) para casi todo ω. En la demostracion de nue-stros resultados usamos otros recientes acerca de los valores singulares de la composicionde operadores y tambien usamos la nocion de angulo de Friedrich entre subespacios deespacios de Hilbert. Ademas, establecemos una condicion analıtica entre A y GΦ(ω) ,que resulta equivalente a la anterior. A pesar de que nosotros estamos interesados en elcaso ` = `(V) (la longitud del espacio invariante por traslaciones enteras V en estudio), lamayorıa de nuestros resultados siguen valiendo cuando `(V) ≤ ` ≤ m.
Por otro lado, incluimos los casos de bases de Riesz y bases ortonormales, que ya sonconocidos.
Recientemente Paternostro [Pat14] obtuvo resultados similares a los probados porBownik y Kaiblinger en [BK06] y a los obtenidos mencionados arriba de [CMP14], peropara espacios invariantes por traslaciones en un contexto mas general que el de L2(Rd).
Este capıtulo esta organizado de la siguiente manera. En la Section 2.2 incluimos lasdefiniciones y resultados que necesitaremos para probar nuestros resultados. En la Section2.3 presentamos los resultados de Bownik y Kaiblinger, mencionados antes, acerca deconjuntos minimales de generadores de espacios invariantes por traslaciones enteras enL2(Rd). Incluimos algunos resultados acerca de los autovalores de matrices conjugadas enla Section 2.4. Estos resultados seran necesarios para las pruebas de nuestros resultados.En la Section 2.5 establecemos nuestros principales resultados con sus pruebas.
2
Linear combinations of generators in systems oftranslates
2.1 Introduction
Let V be a shift invariant space generated by the integer translates of Φ = φ1, . . . , φm.That is, V is the closure of the set of all functions ϕ of the form
ϕ(x) =
m∑j=1
∑k∈Zd
c j,k φ j(x − k), x ∈ Rd, (2.1)
where finitely many c j,k ∈ C are nonzero.
It is interesting to know whether it is possible to obtain a minimal set of generatorsfrom the given generators in Φ. That is whether there exists ψ1, . . . , ψ` ⊆ L2(Rd) with` = length(V) ≤ m such that
S (φ1, . . . , φm) = S (ψ1, . . . , ψ`).
There are many examples with the property that no subset of Φ is a minimal set of gener-ators. So, deleting elements from Φ may not be a successfull procedure.
Concerning this question, Bownik and Kaiblinger in [BK06], showed that a minimalset of generators for V can be obtained from Φ by linear combinations of its elements.Moreover, they proved that almost every set of ` functions that are linear combinations ofφ1, . . . , φm is a minimal set of generators for V (see Section 2.3). We emphasize that thelinear combinations only involve the functions φ1, . . . , φm and not their translations.
In many problems involving shift invariant spaces, it is important that the system oftranslates Tkφ j : k ∈ Zd, j = 1, . . . ,m bears a particular functional analytic structuresuch as being an orthonormal basis, a Riesz basis or a frame. Therefore, it is interestingto know when a minimal set of generators obtained by taking linear combinations ofthe original one has the same structure. More precisely, suppose that Φ = φ1, . . . , φm
generates a shift invariant space V of length `, and assume that a new set of generators
18 Linear combinations of generators in systems of translates
Ψ = ψ1, . . . , ψ` for V is produced by taking linear combinations of the functions in Φ.That is, assume that ψi =
∑mj=1 ai jφ j for i = 1, . . . , ` for some complex scalars ai j. If we
collect the coefficients in a matrix A = ai ji, j ∈ C`×m, then we can write in matrix notation
Ψ = AΦ. We would like to know, which matrices A transfer the structure of Φ over Ψ.Precisely, we study the following question: If we know that Tkφi : k ∈ Zd, i = 1, . . . ,mis a frame for V , when will Tkψi : k ∈ Zd, i = 1, . . . , ` also be a frame for V?
In Section 2.5 we answer this question completely. As we mentioned before, the prop-erty of being a “set of generators” for a SIS V is generically preserved by the actionof a matrix A ([BK06]). This is not anymore valid for the case of frames. This is anunexpected result. More than that, we were able to construct a surprising example ofa shift invariant space V with a set of generators Φ such that their integer translatesTkφi : k ∈ Zd, i = 1, . . . ,m form a frame for V and with the property that no matrixA, of size ` × m with ` < m, transform Φ into a new set of generators that form a framefor V .
Our main result gives exact conditions in order that the frame property is preserved by amatrix A. These conditions are in terms of a particular geometrical relation that has to besatisfied between the nullspace of A and the column space of GΦ(ω) for almost all ω. Theproof uses recent results about singular values of composition of operators and involvesthe Friedrichs angle between subspaces in Hilbert spaces. We also provide an equivalentanalytic condition between A and GΦ(ω) in order that this same result holds. Althoughwe are interested in the case ` = `(V) (the length of the SIS V under study), most of ourresults are still valid when `(V) ≤ ` ≤ m.
For completeness we include the particular case of Riesz bases and orthonormal basesthat are known.
Recently Paternostro [Pat14] obtain similar results to [BK06] and [CMP14] but for shiftinvariant spaces considered in more general contexts than L2(Rd).
This rest of this chapter is organized as follows. In Section 2.2 we set the definitionsand results that we need for the case of L2(Rd). In Section 2.3 we present the results ofBownik and Kaiblinger about minimal generator sets of finitely generated shift invariantspaces in L2(Rd). We include some results about the eigenvalues of conjugated matricesin Section 2.4. In Section 2.5 we state and prove our main results.
2.2 Preliminaries
In this section we only include the notations, definitions and properties that we need forSection 2.3 and Section 2.5, in the context of L2(Rd).
As we mentioned in the introduction, we are interested in when a set of linear com-binations of the generators for a finitely generated SIS inherits some particular structurefrom the original generators. In order to make clear our exposition we use the followingnotation.
2.2 Preliminaries 19
Let Φ = φ1, . . . , φm be a set of functions in L2(Rd). By taking linear combinations ofthe elements of Φ, we construct a new set Ψ = ψ1, . . . , ψ` where ψi =
∑mj=1 ai jφ j and
A = ai ji, j ∈ C`×m is a matrix. Using a matrix notation we set Ψ = AΦ. Then, we consider
the following questions:
Let V = S (Φ) where Φ = φ1, . . . , φm.
• If E(Φ) is a frame for V and `(V) ≤ ` ≤ m, for which matrices A ∈ C`×m, is E(AΦ)a frame for V?
• If E(Φ) is an orthonormal basis (Riesz basis) for V , for which square matrices A, isE(AΦ) an orthonormal basis (Riesz basis) for V?
Sometimes in the chapter by convenient abuse of notation, we will say that a set Φ =
φ1, . . . , φm is a frame for a SIS V to indicate that actually E(Φ) forms a frame for V .
In order to study when E(AΦ) is a frame or a Riesz basis for V we will use Theorem1.5.3. So, we need to know the Gramian associated to AΦ.
Proposition 2.2.1. Let V = S (Φ) be a SIS where Φ = φ1, . . . , φm and let A = ai ji, j ∈
C`×m be a matrix. Consider the set Ψ = ψ1, . . . , ψ` where Ψ = AΦ. Then, the GramianGΨ is a conjugation of GΦ by A, i.e.
GΨ(ω) = AGΦ(ω)A∗
for a.e. ω ∈ [−12 ,
12 ]d.
Proof. Let i, j ∈ 1, · · · , ` be fixed. Then,
[GΨ(ω)]i j =⟨(ψi)ω, (ψ j)ω
⟩=
⟨ m∑k=1
aik(φk)ω,m∑
r=1
a jr(φr)ω⟩
=
m∑k=1
m∑r=1
aika jr
⟨(φk)ω, (φr)ω
⟩=
m∑k=1
m∑r=1
aika jr[GΦ(ω)]kr
= [AGΨ(ω)A∗]i j.
For a matrix B ∈ C`×m we denote by σ(B) the smallest non-zero singular value of B.By Ker(B) and Im(B) we denote the nullspace and column space of B respectively, as anoperator acting by right multiplication, i.e., matrix-vector multiplication.
For a squared positive-semidefinite matrix B such that B = B∗, the eigenvalues andits singular values agree. In particular, σ(B) = λ−(B) where λ−(B) denotes the smallestnon-zero eigenvalue of B.
20 Linear combinations of generators in systems of translates
We need the notion of Friedrichs angle between subspaces that we will use in the fol-lowing sections for stating our results. The Friedrichs angle can be defined for subspacesof a general Hilbert space (see [Deu95, HJ95, Ka84]). However, we will define it forsubspaces of Cn, since this is the context on which we will use it.
Definition 2.2.2. Let N,M , 0 be subspaces of Cn. The Friedrichs angle between Mand N is the angle in [0, π2 ] whose cosine is defined by
G[M,N] = sup|〈x, y〉| : x ∈ M ∩ (M ∩ N)⊥, ‖x‖ = 1, y ∈ N ∩ (M ∩ N)⊥, ‖y‖ = 1.
We define G[M,N] = 0 if M = 0, N = 0, M ⊆ N or N ⊆ M.
As usual, the sine of the Friedrichs angle is defined as F [M,N] =√
1 − G[M,N]2.
We have the following proposition. See [ACRS05] for more details.
Proposition 2.2.3. Let N,M , 0 be closed subspaces of Cn. Then
(i) G[M,N] = G[N,M] = G[M ∩ (M ∩ N)⊥,N] = G[M,N ∩ (M ∩ N)⊥].
(ii) G[M,N] < 1 if and only if M + N is closed.
(iii) G[M,N] = G[M⊥,N⊥].
(iv) G[M,N] = ‖PMPN∩(M∩N)⊥]‖ = ‖PM∩(M∩N)⊥PN‖ = ‖PMPN P(M∩N)⊥‖ = ‖PMPN −
PM∩N‖.
2.3 Minimal generator sets for finitely generated SIS inL2(Rd)
As we mentioned in the introduction, there are examples of sets of generators that donot contain any minimal subset of generators. For instance, consider the shift invariantspace V = S (φ) where φ is such that φ = χ[0,1]. (Here we denote by χM the characteristicfunction of a set M). It can be seen that V = S (φ1, φ2) with φ1, φ2 such that φ1 = χ[0, 1
2 ] andφ2 = χ[ 1
2 ,1]. However, neither φ1 nor φ2 generates V by itself.
An alternative to overcome this problem will be to try to obtain a smaller set of gener-ators, for example a minimal one, considering instead linear combinations of the originalgenerators. Bownik and Kaiblinger in [BK06] gave a solution to this question in this di-rection. Since the new generators ψ1, . . . , ψ` belong to V, they can be approximated in theL2−norm by functions of the form (2.1), i.e., by finite sums of shifts of the original gen-erators. However, they proved that at least one reduced set of generators can be obtainedfrom a linear combination of the original generators without translations. In particular, nolimit or infinite summation is required. Moreover, they proved that almost every set of `functions that are linear combinations of φ1, . . . , φm is a minimal set of generators for V,as the following theorem shows.
2.3 Minimal generator sets for finitely generated SIS in L2(Rd) 21
Theorem 2.3.1. [BK06, Theorem 1] Let V be a finitely generated SIS with length `(V),and Φ = φ1, . . . , φm ⊆ L2(Rd) where `(V) ≤ m and such that V = S (Φ). For every`(V) ≤ ` ≤ m, consider the set of matrices R = A ∈ C`×m : V = S (Ψ),Ψ = AΦ. Then,
(i) C`×m \ R has zero Lebesgue measure.
(ii) N = C`×m \ R can be dense in C`×m.
This result briefly says that given any set of generators of a finitely generated SIS, al-most every matrix (of the right size) transforms it in a new set of generators. In particular,a minimal set of generators can be obtained with this procedure.
As an immediate consequence of Theorem 2.3.1 in case of principal shift invariantspaces we have the following corollary.
Corollary 2.3.2. Let V be a principal SIS, and Φ = φ1, . . . , φm ⊆ L2(Rd) such thatV = S (Φ). Consider the set of vectors R = v ∈ Cm : V = S (Ψ),Ψ = vΦ. Then,
(i) Cm \ R has zero Lebesgue measure.
(ii) N = Cm \ R can be dense in Cm.
The following examples illustrate Corollary 2.3.2 and show that the set N can be asingleton (Example 2.3.3) or indeed be dense in Cm (Example 2.3.4). For a detailedproofs of the following examples we refer the reader to [BK06].
Example 2.3.3. The function sinc ∈ L2(R) is defined by
sinc(t) =sin πtπt
, t ∈ R.
Given m ≥ 1, let φ1, . . . , φm ∈ L2(R) be a collection of distinct translations of the sincfunction. That is,
φ j(t) = sinc(t − t j), t ∈ R, j = 1, . . . ,m,
where t1, . . . , tm ∈ R satisfy t j , tk, for j , k. Let V = S (φ1, . . . , φm).
Then V is the Paley-Wiener space of functions in L2(R) which are band-limited to[−1/2, 1/2], so V is principal. For example, the sinc function itself or any of its shiftsindividually generate V. Indeed any linear combination of the original generators yields asingle generator for V as well, unless all coefficients are zero. Hence, in this example theset N of Theorem 2.3.1 is
N = 0, 0 = (0, . . . , 0) ∈ Cm,
consisting of the zero vector only.
22 Linear combinations of generators in systems of translates
Example 2.3.4. For x ∈ R, let bxc denote the largest integer less or equal x. We define adiscretized version of the Archimedean spiral by γ : [0, 1)→ Z2,
γ(x) = (bu cos 2πuc, bu sin 2πuc), u = tanπ
2x, x ∈ [0, 1).
Next, let
γ0(x) =
γ(x)|γ(x)| if γ(x) , 0,0 otherwise
, x ∈ [0, 1).
Now define φ1, φ2 ∈ L2(R) by their Fourier transforms, obtained from γ0 = (γ01, γ
02) by
φ j(x) =
γ0j (x), x ∈ [0, 1),
0, x ∈ R \ [0, 1), j = 1, 2.
Let V = S (φ1, φ2).
Then V is principal. In fact, the function ψ = λ1φ1 + λ2φ2 is a single generator, V =
S (ψ), if and only if λ1 and λ2 are rationally linearly independent. So here the set N ofTheorem 2.3.1 is
N = (λ1, λ2) ∈ C2 : λ1 and λ2 are rationally linear dependent.
In particular, any rational linear combination of φ1, φ2 fails to generate V. This exampleillustrates Corollary 2.3.2 for the case of real coefficients. Namely, N ∩ R2 is a null setin R2 yet it contains Q2, so it is dense in R2. One can extend this example for the case ofcomplex coefficients.
Since linear combinations of a finite number of functions preserve properties such assmoothness, compact support, bandlimitedness, decay, etc., an interesting consequenceof Bownik and Kaiblinger’s result is that if the generators for V have some additionalproperty, there exists a minimal set of generators that inherits this property.
2.4 Eigenvalues of a conjugated matrix
According to Theorem 1.5.3, we will need to study the eigenvalues of the Gramian GΨ(ω)which is, as Proposition 2.2.1 shows, a conjugation of GΦ(ω) by A. The behaviour of theeigenvalues of the conjugation of a given matrix is in general not very well established.In our case, we will find uniform bounds for the eigenvalues of GΨ(ω).
We now state a recent result that we need for the next section. It is a particular case ofthe result stated in Remark 2.10 in [ACRS05] for square matrices.
Theorem 2.4.1. Let A, B be non zero square matrices in Cm×m. Then,
σ(A)σ(B)F [Ker(A), Im(B)] ≤ σ(AB) ≤ ‖A‖‖B‖ F [Ker(A), Im(B)].
2.4 Eigenvalues of a conjugated matrix 23
In order to adapt Theorem 2.4.1 to our setting we need the following lemma.
Lemma 2.4.2. Let A ∈ C`×m and G ∈ Cm×m with ` ≤ m. If rk(AGA∗) = rk(G) thenKer(AG) = Ker(G).
Proof. Clearly Ker(G) ⊆ Ker(AG). Suppose that dim Ker(G) < dim Ker(AG). Thenrk(AG) < rk(G). Now, rk(AGA∗) ≤ minrk(AG), rk(A) ≤ rk(AG) < rk(G) which is acontradiction. Thus, we must have dim Ker(G) = dim Ker(AG).
Now, from Theorem 2.4.1 and Lemma 2.4.2 we obtain:
Proposition 2.4.3. Let G be a positive-semidefinite matrix in Cm×m such that G = G∗ andA = ai ji, j ∈ C
`×m with ` ≤ m such that rk(G) = rk(AGA∗). Then,
σ(A)2λ−(G)F [Ker(A), Im(G)]2 ≤ λ−(AGA∗) ≤ ‖A‖2‖G‖F [Ker(A), Im(G)].
Proof. Let A ∈ Cm×m be the matrix defined by ai j =
ai j if 1 ≤ i ≤ `0 if ` < i ≤ m
. Then,
AG(A)∗ =( AGA∗ 0
0 0
),
and therefore λ−(AGA∗) = λ−(AGA∗).
Now, using Theorem 2.4.1 and Lemma 2.4.2, we have
λ−(AGA∗) ≥ σ(AG)σ(A∗)F [Ker(AG), Im(A∗)]= σ(AG)σ(A∗)F [Ker(G), Im(A∗)]≥ σ(A)λ−(G)F [Ker(A), Im(G)]σ(A∗)F [Ker(G), Im(A∗)],
and
λ−(AGA∗) ≤ ‖AG‖‖A∗‖F [Ker(AG), Im(A∗)] ≤ ‖A‖2‖G‖F [Ker(G), Im(A∗)].
By the properties of the sine of the Friedrichs angle it can be seen thatF [Ker(G), Im(A∗)] = F [Ker(A), Im(G)]. Using that σ(A) = σ(A) = σ(A∗) and ‖A‖ =
‖A‖, we finally obtain
σ(A)2λ−(G)F [Ker(A), Im(G)]2 ≤ λ−(AGA∗) ≤ ‖A‖2‖G‖F [Ker(A), Im(G)].
We finish the proof by observing that Ker(A) = Ker(A).
The last result of this section gives an equivalent condition for rk(AGA∗) = rk(G) tohold, and we will use it in the next section.
Lemma 2.4.4. Let A ∈ C`×m with ` ≤ m and G ∈ Cm×m such that G is positive-semidefiniteand G = G∗. Then, rk(AGA∗) = rk(G) if and only if Ker(A) ∩ Im(G) = 0.
24 Linear combinations of generators in systems of translates
Proof. Note that, since G is positive semidefinite, and G = G∗, it is always truethat rk(AGA∗) = rk(AG1/2G1/2A∗) = rk(AG1/2). Then, rk(AGA∗) = rk(G) if andonly if dim(Ker(AG1/2)) = dim(Ker(G)). Thus, since Im(G) = Im(G1/2) and thenKer(G) = Ker(G1/2), we want to prove that dim(Ker(AG1/2)) = dim(Ker(G1/2) if andonly if Ker(A) ∩ Im(G1/2) = 0. Now, using that Ker(G1/2) ⊆ Ker(AG1/2), we havedim(Ker(AG1/2)) = dim(Ker(G1/2) if and only if Ker(AG1/2) = Ker(G1/2).
Finally, it is easy to check that the condition Ker(AG1/2) = Ker(G1/2) is equivalent toKer(A) ∩ Im(G) = 0.
2.5 Linear combinations of frame generators in systemsof translates in L2(Rd)
As we mentioned in Section 2.3, Bownik and Kaiblinger proved that given any set ofgenerators of a finitely generated SIS, almost every matrix (of a right size) transforms itin a minimal set of generators.
An interesting question arises here. When is the set of generators obtained, a set offrame generators? (i.e. the integer translations form a frame of the SIS?) More preciselywe want to obtain new sets of generators of the form Ψ = AΦ with the additional propertyof being a frame.
First, we show that in order to E(Ψ) to be a frame, E(Φ) needs to be a frame. Moreprecisely we prove:
Proposition 2.5.1. Let Φ = φ1, . . . , φm ⊆ L2(Rd) be a generator set for a SIS V of length`(V) ≤ m and suppose that E(Φ) is a Bessel sequence but not a frame for V. Then, foreach matrix A ∈ C`×m, with `(V) ≤ ` ≤ m, E(Ψ) is not a frame for V where Ψ = AΦ.
After this result, the right question will be which matrices (of the right size) map framegenerator sets into new frame generator sets? The answer of this question is not as directas is the case of a plain set of generators and it will take us the rest of the section. Let usfirst start with the proof of Proposition 2.5.1
Proof of Proposition 2.5.1. Let A ∈ C`×m. If Ψ = AΦ is not a generator set for V , then Ψ
it is not a frame for V . Thus, suppose that Ψ is a set of generators for V .
Since E(Φ) is a Bessel sequence but not a frame for V , the lower frame inequality in(1.3) is not satisfied. Therefore, there exists fnn∈N ⊆ V such that
tn(Φ) :=m∑
j=1
∑k∈Zd
|〈 fn,Tkφ j〉|2 → 0, when n→ +∞.
2.5 Linear combinations of frame generators in systems of translates in L2(Rd) 25
Now,
tn(Ψ) =∑i=1
∑k∈Zd
|〈 fn,Tkψi〉|2
=∑i=1
∑k∈Zd
|
m∑j=1
ai j〈 fn,Tkφ j〉|2
≤∑i=1
∑k∈Zd
m∑j=1
|ai j|2
m∑
j=1
|〈 fn,Tkφ j〉|2
=
∑i=1
m∑j=1
|ai j|2
tn(Φ).
Then, tn(Ψ) → 0, when n → +∞ and thus, E(Ψ) does not satisfy the lower frame in-equality.
When E(Φ) is not a Bessel sequence it can happen that for certain matrix A, E(AΦ)is a Bessel sequence. To construct an easy example take φ ∈ L2(R) such that E(φ) is aframe for S (φ). Now, choose a second generator φ ∈ S (φ) such that E(φ) is not a Besselsequence. Thus, if Φ = φ, φ, S (Φ) = S (φ) and E(φ, φ) is not a Bessel sequence.However, taking A = (1, 0), we get that E(AΦ) is a Bessel sequence. The hypothesis inthe above proposition of E(Φ) being a Bessel sequence it is not very restrictive and greatlysimplifies the treatment.
We now point out a property about the result of Bownik and Kaiblinger that will beimportant in what follows.
From the proof of Theorem 2.3.1, it follows that the set R can be described in terms ofthe Gramian as
R = A ∈ C`×m : rk(GΦ(ω)) = rk(AGΦ(ω)A∗) for a.e. ω ∈ [−1/2, 1/2]d, (2.2)
where ` is a number between length of S (Φ) and m. Now, using Lemma 2.4.4
R = A ∈ C`×m : Ker(A) ∩ Im(GΦ(ω)) = 0 for a.e. ω ∈ [−1/2, 1/2]d.
Remark 2.5.2. When ` is exactly the length of S (Φ), note that if A ∈ R, by (1.8), ` = rk(A)and then A is full rank.
As we have already discussed, a key point in our problem is the behavior of the eigen-values of conjugated matrices. Here one wants to get a smaller set of generators froma given large set of generators. In terms of matrices, this translates in conjugating theGramian by rectangular matrices. The behavior of the eigenvalues in this case is notvery well understood. However we are able to exactly determine those matrices thatyield frames, in terms of the Friedrichs angle, and using recent results by Antezana et al[ACRS05] on singular values of composition of operators.
The main result of this section is the following theorem:
26 Linear combinations of generators in systems of translates
Theorem 2.5.3. Let Φ = φ1, . . . , φm ⊆ L2(Rd) such that E(Φ) is a frame for V = S (Φ)and suppose that `(V) ≤ ` ≤ m. Let A ∈ C`×m be a matrix and consider Ψ = ψ1, . . . , ψ`where Ψ = AΦ. Then, E(Ψ) is a frame for V if and only if A satisfies the following twoconditions
1. A ∈ R where R is as in (2.2).
2. There exists δ > 0 such that F [Ker(A), Im(GΦ(ω))] ≥ δ for a.e. ω ∈ [−12 ,
12 ]d.
Proof of Theorem 2.5.3. Let 0 < α ≤ β be the frame bounds for E(Φ).
First suppose that E(Ψ) is a frame for V and let β′ ≥ α′ > 0 be its frame bounds. Since,in particular, Ψ is a generator set for V, A belongs to R.
By Proposition 2.4.3, we have that
λ−(GΨ(ω)) = λ−(AGΦ(ω)A∗) ≤ ‖A‖2‖GΦ(ω)‖ F [Ker(A), Im(GΦ(ω))],
for a.e. ω ∈ [−12 ,
12 ]d.
Using Theorem 1.5.3, α′ ≤ λ−(GΨ(ω)) and ‖GΦ(ω)‖ ≤ β. Thus,
α′ ≤ ‖A‖2βF [Ker(A), Im(GΦ(ω))].
Then, item (2) is satisfied taking δ = α′
‖A‖2β .
Conversely. Since A ∈ R, V = S (Ψ) and rk(AGΦ(ω)A∗) = rk(GΦ(ω)) for a.e. ω ∈[−1
2 ,12 ]d.
Now, for a.e. ω ∈ [−12 ,
12 ]d, we apply the lower inequality that Proposition 2.4.3 gives
to GΦ(ω) and A. Then,
λ−(AGΦ(ω)A∗) ≥ σ(A)2λ−(GΦ(ω))F [Ker(A), Im(GΦ(ω))]2 ≥ σ(A)2αδ2.
On the other hand, by Theorem 1.5.3,
‖GΨ(ω)‖ = ‖AGΦ(ω)A∗‖ ≤ ‖A‖2‖GΦ(ω)‖ ≤ ‖A‖2β
and from this it follows that the eigenvalues of GΨ(ω) are bounded above by ‖A‖2β.
Therefore, Σ(GΨ(ω)) ⊆[σ(A)2αδ2, ‖A‖2β
]∪ 0 for a.e. ω ∈ [−1
2 ,12 ]d and the result
follows from Theorem 1.5.3.
As a consequence of the above theorem, we have the following result. We imposemore restrictive condition on GΦ(ω) than in Theorem 2.5.3. However the new hypothesisis easy to check and avoid the calculation of the sine of the Friedrichs angle.
Corollary 2.5.4. Let Φ = φ1, . . . , φm ⊆ L2(Rd) such that E(Φ) is a frame for V = S (Φ)and suppose that `(V) ≤ ` ≤ m. Consider A ∈ C`×m such that Ker(A) = Ker(GΦ(ω))for a.e. ω ∈ [−1
2 ,12 ]d, and dim(Ker(A)) = m − `. If A ∈ R where R is as in (2.2), and
Ψ = ψ1, . . . , ψ` where Ψ = AΦ, then E(Ψ) is a frame for V.
2.5 Linear combinations of frame generators in systems of translates in L2(Rd) 27
Proof. Since Ker(A) ⊆ Ker(GΦ(ω)) and Ker(GΦ(ω)) = Im(GΦ(ω))⊥, it follows thatF [Ker(A), Im(GΦ(ω))] = 1.
Example 2.5.5. Let V = S (Φ) be the shift invariant space generated by Φ = φ1, φ2, φ3 ⊆
L2(R), where φ1, φ2, φ3 are defined by
φ1(ω) = −8χ[− 12 ,
12 ](ω) + 4χ[ 1
2 ,32 ](ω),
φ2(ω) = χ[− 12 ,
12 ](ω) + 4χ[ 3
2 ,52 ](ω),
φ3(ω) = χ[ 12 ,
32 ](ω) + 8χ[ 3
2 ,52 ](ω).
Then, the associated Gramian is
GΦ(ω) =
80 −8 4−8 17 324 32 65
.It can be seen that G2
Φ(ω) = 81GΦ(ω) and rk(GΦ(w)) = 2 for all ω ∈ [−1
2 ,12 ]. Thus, by
Theorem 1.5.3, E(Φ) is a frame for V and using that
`(V) = esssupω∈[− 1
2 ,12 ]d
rk(GΦ(ω)),
we obtain `(V) = 2.
On the other side, Ker(GΦ(ω)) = span(1, 8,−4). Then, V satisfy the hypothesis ofCorollary 2.5.4. In what follows, we construct all possible matrices A ∈ C2×3 such thatKer(A) = span(1, 8,−4) and A ∈ R. These matrices give frames E(Ψ) with Ψ = AΦ.
Since Ker(A) = span(1, 8,−4), A has the form
A =
((4b − 8a) a b(4d − 8c) c d
),
with a, b, c, d ∈ C.
Now, A ∈ R if and only if rk(AGΦ(ω)A∗) = 2. Now rk(AGΦ(ω)A∗) = 2 if and only ifdet(AGΦ(ω)A∗) , 0. Since det(AGΦ(ω)A∗) = (81)3(ad − bc)2, we conclude that A ∈ R ifand only if ad − bc , 0.
Condition (2) in Theorem 2.5.3 is a geometric property that A and GΦ need to satisfyin order to A preserves the frame property of E(Φ) over E(Ψ). We now state a result onwhich we give an analytic way to express conditions (1) and (2) of Theorem 2.5.3. In thiscase, ` will be exactly the length of the SIS.
Theorem 2.5.6. Let Φ = φ1, . . . , φm ⊆ L2(Rd) such that E(Φ) is a frame for V = S (Φ)and suppose that `(V) = ` ≤ m. Let A ∈ C`×m be a matrix and consider Ψ = ψ1, . . . , ψ`where Ψ = AΦ. Then, E(Ψ) is a frame for V if and only if the following condition betweenA and GΦ is satisfied: AA∗ is invertible and
esssupω∈[− 1
2 ,12 ]d
‖(Im − A∗(AA∗)−1A)GΦ(ω)G†Φ
(ω)‖ < 1. (2.3)
Here, Im is the identity in Cm×m and G†Φ
(ω) is the Moore-Penrose pseudoinverse of GΦ(ω).
28 Linear combinations of generators in systems of translates
Proof. Let us first prove that conditions (1) and (2) of Theorem 2.5.3 imply condition(2.3).
Note that condition (2) is equivalent to G[Ker(A), Im(GΦ(ω))] ≤√
1 − δ2 for a.e. ω ∈[−1
2 ,12 ]d. Now, since Ker(A) ∩ Im(GΦ(ω)) = 0 for a.e ω ∈ [−1
2 ,12 ]d, it follows from
Proposition 2.2 in [ACRS05] that
G[Ker(A), Im(GΦ(ω))] = ‖PKer(A)PIm(GΦ(ω))‖ for a.e. ω ∈ [−12,
12
]d,
where PKer(A) and PIm(GΦ(ω)) denote the orthogonal projection onto Ker(A) and Im(GΦ(ω)respectively. Using the Moore-Penrose pseudoinverse we can write
‖PKer(A)PIm(GΦ(ω))‖ = ‖(Im − A†A)GΦ(ω)G†Φ
(ω)‖.
Finally, since A is full rank, we replace A† = A∗(AA∗)−1 and then the assertion follows.
Conversely. Suppose that (2.3) holds. Then, we have that
‖PKer(A)PIm(GΦ(ω))‖ ≤ γ < 1 for all ω ∈ [−12,
12
]d \ Z,
where Z is a set with Lebesgue measure zero and
γ = esssupω∈[− 1
2 ,12 ]d
‖(Im − A∗(AA∗)−1A)GΦ(ω)G†Φ
(ω)‖.
Fix ω ∈ [−12 ,
12 ]d \ Z and suppose there exists x ∈ Ker(A) ∩ Im(GΦ(ω)) with ‖x‖ = 1.
Then, ‖PKer(A)PIm(GΦ(ω))x‖ = ‖x‖ = 1 which is a contradiction. Therefore,
Ker(A) ∩ Im(GΦ(ω)) = 0 for all ω ∈ [−12,
12
]d \ Z,
and this gives that (1) in Theorem 2.5.3 is satisfied. Having Ker(A)∩ Im(GΦ(ω)) = 0 fora.e. ω ∈ [−1
2 ,12 ]d, condition (2) of Theorem 2.5.3 can be obtained with similar arguments
as in the first part of this proof.
Now, we consider the case of Riesz bases. We obtain necessary and sufficient condi-tions on A in order to preserve Riesz bases of translates. Different from the frame case,the conditions on A do not depend on the shift invariant space.
First, observe that, if Φ = φ1, . . . , φm ⊆ L2(Rd) is such that E(Φ) is a Riesz (orthonor-mal) basis for V = S (Φ), by Theorem 1.5.3, GΦ(ω) is an m × m invertible matrix for a.e.ω ∈ [−1
2 ,12 ]d. Thus, using that
`(V) = esssupω∈[− 1
2 ,12 ]d
rk(GΦ(ω)),
we have that `(V) = m. Therefore, in order to preserve Riesz (orthonormal) bases, weneed to consider squared matrices A ∈ Cm×m.
2.5 Linear combinations of frame generators in systems of translates in L2(Rd) 29
Proposition 2.5.7. Let Φ = φ1, . . . , φm ⊆ L2(Rd) such that E(Φ) is a Riesz basis forV = S (Φ). Let A ∈ Cm×m be a matrix and consider Ψ = ψ1, . . . , ψm with Ψ = AΦ. Then,E(Ψ) is a Riesz basis for V if and only if A is an invertible matrix.
Proof. Let 0 < α ≤ β be the Riesz bounds for E(Φ). Suppose first that A is invertible.Then, for a.e. ω ∈ [−1
2 ,12 ]d,
‖GΨ(ω)‖ = ‖AGΦ(ω)A∗‖ ≤ ‖A‖2‖GΦ(ω)‖ ≤ ‖A‖2β,
‖(GΨ(ω))−1‖ ≤ ‖A−1‖2‖(GΦ(ω))−1‖ ≤ ‖A−1‖21α.
Therefore, by Theorem 1.5.3, E(Ψ) is a Riesz basis for V .
Conversely, if E(Ψ) is a Riesz basis for V, it follows from Theorem 1.5.3 that GΨ(ω) isinvertible for a.e. ω ∈ [−1
2 ,12 ]d. Since GΦ(ω) is invertible as well for a.e. ω ∈ [−1
2 ,12 ]d we
have that A is invertible.
Remark 2.5.8. The above result shows that every invertible matrix preserves Riesz basesof translates. On the other side, it is known that the set A ∈ Cm×m : det(A) = 0 hasLebesgue measure zero. Thus, we have that almost every matrix (exactly those that areinvertible) preserves Riesz bases of translates. We can then connect this with Bownik andKaiblinger’s result as follows. If in addition to the hypothesis of Theorem 2.3.1 we ask theRiesz basis condition on E(Φ), we have that the set R is equal to A ∈ Cm×m : det(A) , 0.
It is worth to mention that Proposition 2.5.7 can be proven as a corollary of Theorem2.5.3 or independently from this result using Ostrowski’s Theorem:
Theorem 2.5.9. [HJ90, Theorem 4.5.9] Let A, S ∈ Cm×m with A Hermitian and S non-singular. Let the eigenvalues of A and S S ∗ be arranged in increasing order. For eachk = 1, 2, . . . , there exists a positive real number θk such that λ1(S S ∗) ≤ θk ≤ λn(S S ∗) and
λk(S AS ∗) = θkλk(A).
Applying this result to G = GΦ(ω) for a.e. ω ∈ [−12 ,
12 ]d, uniform bounds for the
eigenvalues of AGΦ(ω)A∗ can be found.
For frames, in the special case when the initial set of generators has exactly `(V) ele-ments, we have that every invertible matrix yields a set of generators that is a frame forV. This is stated in the next result and its proof is analogous to the proof of Proposition2.5.7. It can be also viewed as a corollary of Theorem 2.5.6.
Theorem 2.5.10. Let Φ = φ1, . . . , φ` ⊆ L2(Rd) such that E(Φ) is a frame for V = S (Φ)and suppose that `(V) = `. Let A ∈ C`×` be a matrix and consider Ψ = ψ1, . . . , ψ` whereΨ = AΦ. Then, E(Ψ) is a frame for V if and only if A is an invertible matrix.
Finally, in case of orthonormal bases, we have:
Proposition 2.5.11. Let Φ = φ1, . . . , φm ⊆ L2(Rd) such that E(Φ) is an orthonormalbasis for V = S (Φ) and let A ∈ Cm×m be a matrix. Consider Ψ = AΦ. Then, E(Ψ) is anorthonormal basis for V if and only if A is an unitary matrix.
30 Linear combinations of generators in systems of translates
Proof. Note that if A = ai ji, j then,
〈Tkψi,Tk′ψi′〉 =
m∑j, j′
ai jai′ j′〈Tkφ j,Tk′φ j′〉 = (AA∗)ii′δ(k − k′),
and from here it follows that E(Ψ) is an orthonormal set if and only if A is unitary.
For the completeness of E(Ψ) on V we use that, since A is unitary and Ψ = AΦ, wecan write Φ = A∗Ψ. Then Tkφ j =
∑mi=1 ai jTkψi, k ∈ Z, j = 1, . . . ,m and the result
follows.
Theorem 2.5.7 shows that almost every square matrix maps Riesz bases generatorsin Riesz bases generators. For the case of frames it might happen that condition (2) ofTheorem 2.5.3 is not satisfied for any matrix A. That is exactly what we show in thefollowing example on which we present a finitely generated SIS for which any linearcombination of its generators yields to a minimal set of generators that is not a frame oftranslates.
Example 2.5.12. Let φ1, φ2 ∈ L2(R2) defined by
φ1(ω1, ω2) = − sin(2πω1)χ[− 12 ,
12 ]2(ω1, ω2)
andφ2(ω1, ω2) = e2πiω2 cos(2πω1)χ[− 1
2 ,12 ]2(ω1, ω2).
Consider the shift invariant space generated by Φ = φ1, φ2, V = S (φ1, φ2). We willsee that E(φ1, φ2) is a frame for V , that V is a principal SIS and that E(AΦ) is not a framefor V for any matrix A ∈ C1×2.
We first compute GΦ(ω1, ω2). Let (ω1, ω2) ∈ [−12 ,
12 ]2. Then, we have
GΦ(ω1, ω2) =
(sin2(2πω1) −e−2πiω2 sin(2πω1) cos(2πω1)
−e2πiω2 sin(2πω1) cos(2πω1) cos2(2πω1)
).
Note that GΦ(ω1, ω2) = G2Φ
(ω1, ω2) for all (ω1, ω2) ∈ [−12 ,
12 ]2. Thus, by Theorem 1.5.3,
E(φ1, φ2) is a frame for V. Further, it can be seen that rk(GΦ(ω1, ω2)) = 1 for a.e. (ω1, ω2) ∈[−1
2 ,12 ]2 and then V has length 1. Moreover, V is the Paley-Wiener space
PW = f ∈ L2(R2) : supp( f ) ⊆ [−1/2, 1/2]2.
Let A ∈ C1×2. Without loss of generality we can suppose that A = (a1 a2) with |a1|2 +
|a2|2 = 1. Then, A can be written as A = (cos(θ)e2πiβ sin(θ)e2πiβ′) for θ ∈ [0, π2 ] and
β, β′ ∈ R.
Therefore, the Gramian associated to AΦ is
AGΦ(ω1, ω2)A∗ = sin2(2πω1) cos2(θ) + cos2(2πω1) sin2(θ)−2 cos(2π(w2 − β
′ + β)) sin(2πω1) cos(2πω1) sin(θ) cos(θ).
2.5 Linear combinations of frame generators in systems of translates in L2(Rd) 31
Observe that for each θ, β and β′ fixed,
AGΦ(ω1, ω2)A∗ , 0 for a.e. (ω1, ω2) ∈ [−1/2, 1/2]2.
In particular,
rk(GΦ(ω1, ω2)) = rk(AGΦ(ω1, ω2)A∗) for a.e. (ω1, ω2) ∈ [−1/2, 1/2]2
and then, every matrix A preserves generators.
Let ω2 ∈ [−1/2, 1/2] such that ω2 = β′ − β + k for some k ∈ Z. Then,
AGΦ(ω1, ω2)A∗ = sin2(2πω1) cos2(θ) + cos2(2πω1) sin2(θ)−2 sin(2πω1) cos(2πω1) sin(θ) cos(θ)
= sin2(2πω1 − θ).
Now, taking ω1 = θ2π , we get AGΦ(ω1, ω2)A∗ = 0. Then, since the Gramian associated
to AΦ is a continuos function with a zero, condition (b) in item (3) of Theorem 1.5.3 cannever be fullfiled. Thus, E(AΦ) can not be a frame for V .
Capıtulo 3:Extra invariancia en espacios invariantes por
traslaciones enteras(Resumen)
En este capıtulo repasaremos resultados ya conocidos acerca de la caracterizacion de losespacios invariantes por traslaciones enteras que no solo son invariantes bajo estas trasla-ciones si no que presentan ademas invariancia sobre un conjunto particular de traslacionesen Rd.
Para un espacio invariante por traslaciones enteras (EITE) V ⊆ L2(Rd), definimos suconjunto de invariancia de la siguiente manera
MV = x ∈ Rd : Tx f ∈ V,∀ f ∈ V.
Diremos que un subespacio cerrado V ⊆ L2(Rd) es un espacio M−invariante si Tm f ∈ Vpara todo m ∈ M y para todo f ∈ V.
Un caso lımite resulta cuando M = Rd. En este caso diremos que V es un espacioinvariante por cualquier traslacion real. Un ejemplo en R de este tipo de espacios es elespacio de Paley-Wiener de funciones que son de banda limitada en [−1/2, 1/2] definidode la siguiente forma
PW = f ∈ L2(R) : sop ( f ) ⊆ [−1/2, 1/2].
Este espacio resulta invariante bajo cualquier traslacion real pues si f ∈ PW y x ∈ R,tenemos que
sop (tx f ) = sop (e−2πix f ) = sop ( f ).
Luego, tx f ∈ PW para todo x ∈ R.
Es facil probar que, para un espacio medible Ω ⊆ Rd, los espacios
VΩ := f ∈ L2(Rd) : sop( f ) ⊆ Ω (2.4)
34 Resumen del Capıtulo 3
resultan invariantes bajo cualquier traslacion real. Mas aun, el Teorema de Wiener (ver[Hel64]) prueba que cualquier espacio cerrado invariante bajo cualquier traslacion real deL2(Rd) es de la forma (2.4).
Por otro lado, existen espacios invariantes por traslaciones enteras que solo son invari-antes bajo traslaciones enteras. Por ejemplo, es facil ver que el siguiente espacio principalinvariante por traslaciones enteras es solo invariante por dichas traslaciones:
V = S (χ[0,1)) = span tkχ[0,1) : k ∈ Z.
Sea V un espacio invariante por traslaciones enteras de longitud ` y sea M un sub-grupo aditivo de Rd que contiene a Zd. Diremos que V tiene extra invariancia M si V esM−invariante. Notemos que en este caso, si Φ es un conjunto de generadores de V , i.e.V = S (Φ), entonces
S (Φ) = spanTkφ : φ ∈ Φ, k ∈ Zd = spanTαφ : φ ∈ Φ, α ∈ M.
Es interesante analizar como es la estructura del conjunto de invariancia MV para unEITE dado V ⊆ L2(Rd). Siguiendo esta direccion, en [ACHKM10] los autores caracter-izan aquellos espacios invariantes por traslaciones enteras V ⊆ L2(R) que tienen extrainvariancia. Ellos muestran que si V es un espacio invariante por traslaciones enteras,entonces su conjunto de invariancia es un subgrupo cerrado de R que contiene a Z. Comoconsecuencia de este resultado, como cualquier subgrupo aditivo de R es o bien discretoo denso, entonces se tiene que o V resulta invariante bajo cualquier traslacion real o existeun maximo numero entero n tal que V es 1
nZ−invariante. Ademas, en [ACHKM10], losautores establecen distintas caracterizaciones acerca de cuando un espacio invariante portraslaciones enteras resulta ademas 1
nZ−invariante.
Generalizando los resultados anteriores, en [ACP11] los autores consideran el caso d-dimensional. Ellos caracterizan la extra invariancia del EITE V para el caso donde MV noes todo Rd. La mayor diferencia con el caso uno dimensional es que existen subgrupos enRd que no son ni discretos ni densos. Luego, no se obtiene en forma directa que todas lascaracterizaciones dadas para R en [ACHKM10] sigan siendo validas en varias variables.
Por otro lado, en [ACP10], los autores consideran otra generalizacion del problemaanterior. Mas precisamente, ellos consideran el mismo problema pero en el contexto deespacios invariantes por traslaciones en grupos localmente compactos y abelianos (LCA).
El capıtulo esta organizado de la siguiente manera. En la Section 3.2 repasamos laestructura del conjunto de invariancia para un espacio invariante por traslaciones en-teras y ademas incluimos algunos resultados acerca de los subgrupos cerrados aditivosen Rd. Por otro lado, presentamos resultados concernientes a la estructura de los es-pacios M−invariantes principales. Repasamos algunas caracterizaciones realizadas por[ACP11], acerca de la extra invariancia de espacios invariantes bajo traslaciones enterasde L2(Rd) en la Section 3.3.
3
Extra invariance of shift invariant spaces
3.1 Introduction
In this chapter we review known results about the characterization of shift invariant spacesthat are not only invariant under integer translations, but are also invariant under someparticular set of translations of Rd.
For a shift invariant space, V ⊆ L2(Rd) we can define its invariance set as follows
MV = x ∈ Rd : Tx f ∈ V,∀ f ∈ V.
We will say that a closed subspace V ⊆ L2(Rd) is M−invariant if Tm f ∈ V for all m ∈ Mand for all f ∈ V.
A limit case is when M = Rd. In this case we will say that V is a translation invariantsubspace. One example of a translation invariant space in R is the Paley-Wiener space offunctions that are bandlimited to [−1/2, 1/2] defined by
PW := f ∈ L2(R) : supp ( f ) ⊆ [−1/2, 1/2].
This space is a translation invariant space since if f ∈ PW and x ∈ R, we have thatsupp (Tx f ) = supp (e−2πix f ) = supp ( f ), thus Tx f ∈ PW for all x ∈ R.
It is easy to prove that for a measurable set Ω ⊆ Rd, the space
VΩ := f ∈ L2(Rd) : supp( f ) ⊆ Ω (3.1)
is translation invariant. Moreover, Wiener’s theorem (see [Hel64]) proves that any closedtranslation invariant subspace of L2(Rd) is of the form (3.1).
On the other hand, there exist shift invariant spaces that are only invariant under integertranslates. For example, it is easy to see that the following principal shift invariant spaceis only invariant under integer translates:
V = S (χ[0,1)) = span Tkχ[0,1) : k ∈ Z.
36 Extra invariance of shift invariant spaces
If V is a shift invariant space of length ` and M is an additive subgroup of Rd containingZd, we will say that V has extra invariance M if V is M−invariant. Note that in this case,if Φ is a set of generators of V , i.e. V = S (Φ), then
S (Φ) = spanTkφ : φ ∈ Φ, k ∈ Zd = spanTαφ : φ ∈ Φ, α ∈ M.
One interesting question is how is the structure of the invariance set MV for a SISV ⊆ L2(Rd) given. In this direction, in [ACHKM10] the authors characterize those shiftinvariant spaces V ⊆ L2(R) that have extra invariance. They showed that if V is a shiftinvariant space then the invariance set is a closed additive subgroup of R containing Z.As a consequence, since every additive subgroup of R is either discrete or dense, theneither V is translation invariant, or there exists a maximum positive integer n such thatV is 1
nZ−invariant. They established different characterizations of when a shift invariantspace is 1
nZ− invariant.
In [ACP11] the authors consider the d-dimensional case. They characterize the extrainvariance of V when M is not all Rd. The main difference with the one dimensionalcase is that there are subgroups of Rd that are neither discrete or dense. So, it is notstraightforward that all the characterizations given in [ACHKM10] are still valid in severalvariables.
The chapter is organized as follows. In Section 3.2 we resume the structure of theinvariance set and the closed additive subgroups in Rd. Also, we present a result aboutthe structure of principal M−invariant spaces. We present characterizations for the extrainvariance of shift invariant spaces in L2(Rd) in Section 3.3.
3.2 Preliminaries
In this section we present some known results concerning the structure of the invarianceset of a shift invariant space in Rd and the structure of the additive closed subgroups of Rd
containing Zd. We will not go into details. We refer the reader to [Bou74, ACP11] and thereferences therein. Also, we present results about the structure of principal M−invariantspaces (see [ACP11] for details).
Definition 3.2.1. Let V ⊆ L2(Rd) be a shift invariant space. We define the invariance setof V by
MV = x ∈ Rd : Tx f ∈ V,∀ f ∈ V. (3.2)
If Φ is a set of generators for V, it is easy to check that
MV = x ∈ Rd : Txϕ ∈ V,∀ϕ ∈ Φ.
Proposition 3.2.2. Let V be a SIS of L2(Rd) and let MV be defined as in (3.2). Then MV
is an additive closed subgroup of Rd containing Zd.
3.2 Preliminaries 37
Then, we are interested in closed subgroups of Rd containing Zd. For their understand-ing, the following definition is important.
Definition 3.2.3. Let M be a subgroup of Rd. Consider the set
M∗ = x ∈ Rd : 〈x,m〉 ∈ Z, ∀m ∈ M.
Then M∗ is a subgroup of Rd. In particular, (Zd)∗ = Zd.
Also, we give the definition of range and dimension of a closed subgroup of Rd.
Definition 3.2.4. Given M a subgroup of Rd, the range of M, denoted by r(M), is thedimension of the subspace generated by M as a real vector space.
Definition 3.2.5. Given M a closed subgroup of Rd, there exists a subspace whose dimen-sion is the largest of the dimensions of all the subspaces contained in M. We will denoteby d(M) the dimension of V. Note that d(M) can be zero.
Remark 3.2.6. Given M a subgroup of Rd,
0 ≤ d(M) ≤ r(M) ≤ d.
Now we list some properties.
Proposition 3.2.7. Let M,N be subgroups of Rd.
(a) M∗ is a closed subgroup of Rd.
(b) If N ⊆ M, then M∗ ⊆ N∗.
(c) If M is closed, then r(M∗) = d − d(M) and d(M∗) = d − r(M).
(d) (M∗)∗ = M.
Let H be a subgroup of Zd with r(H) = q, we will say that a set v1, . . . , vq ⊆ H is abasis for H if for every x ∈ H there exist unique k1, . . . , kq ∈ Z such that
x =
q∑i=1
kivi.
Observe that v1, . . . , vd ⊆ Zd is a basis for Zd if and only if the determinant of the matrix
A which has v1, . . . , vq as columns is 1 or −1.
Given B = v1, . . . , vd a basis for Zd, we call B = w1, . . . ,wd a dual basis for B if〈vi,w j〉 for all 1 ≤ i, j ≤ d.
The following result gives a characterization of closed subgroups of Rd containing Zd.For the proof see [Bou74].
38 Extra invariance of shift invariant spaces
Theorem 3.2.8. Let M ⊆ Rd. The following conditions are equivalent:
(a) M is a closed subgroup of Rd containing Zd and d(M) = d − q.
(b) There exist a basis v1, . . . , vd for Zd and integers a1, . . . , aq satisfyingai+1 ≡ 0 (mod ai) for all 1 ≤ i ≤ q − 1, such that
M =
q∑
i=1
ki1ai
vi +
d∑j=q+1
t jv j : ki ∈ Z, t j ∈ R
.Furthermore, the integers q and a1, . . . , aq are uniquely determined by M.
Remark 3.2.9. If v1, . . . , vd and a1, . . . , aq are as in Theorem 3.2.8, let us define the lineartransformation T as
T : Rd → Rd, T (ei) = vi, ∀ 1 ≤ i ≤ d.
Then T is an invertible transformation that satisfies
M = T(
1a1Z × · · · ×
1aqZ × Rd−q
).
If w1, . . . ,wd is the dual basis for v1, . . . , vd, the inverse of the adjoint of T is definedby
(T ∗)−1 : Rd → Rd, (T ∗)−1(ei) = wi, ∀ 1 ≤ i ≤ d.
ThereforeM∗ = (T ∗)−1(a1Z × · · · × aqZ × 0d−q).
From now on, M will be a closed subgroup of Rd containing Zd and M∗ the groupdefined as in Definition 3.2.3.
We will denote for a set of functions Φ ⊆ L2(Rd), Φ = ϕ : ϕ ∈ Φ.
Definition 3.2.10. We will say that a closed subspace V ⊆ L2(Rd) is M−invariant iftm f ∈ V for all m ∈ M and f ∈ V.
Given Φ ⊆ L2(Rd), the M−invariant space generated by Φ is
S M(Φ) = span tmϕ : m ∈ M, ϕ ∈ Φ.
If Φ = ϕ we write S M(ϕ) instead of S M(ϕ) and we say that S M(ϕ) is a principalM−invariant space. When M = Zd, we will simply write S (ϕ) instead of S Zd (ϕ).
Principal shift invariant spaces have been completely characterized by [dBDR94,RS95] as follows.
3.3 Characterization of extra invariance of SIS in L2(Rd) 39
Theorem 3.2.11. Let f ∈ L2(Rd) be given. If g ∈ S ( f ), then there exists a Zd−periodicfunction η such that g = η f .
Conversely, if η is a Zd−periodic function such that η f ∈ L2(Rd), then the function gdefined by g = η f belong to S ( f ).
In [ACP11], the authors generalize the previous theorem to the M−invariant case. Incase that M is discrete, Theorem 3.2.11 follows easily by rescaling. The difficulty ariseswhen M is not discrete. Theorem 3.2.11 was proved in [dBDR94] for the lattice case. In[ACP11], the authors adapt their arguments to prove the following result.
Theorem 3.2.12. Let f ∈ L2(Rd) and M a closed subgroup of Rd containing Zd. If g ∈S M( f ), then there exists an M∗−periodic function η such that g = η f .
Conversely, if η is an M∗−periodic function such that η f ∈ L2(Rd), then the function gdefined by g = η f belong to S M( f ).
3.3 Characterization of extra invariance of SIS in L2(Rd)
Given M a closed subgroup of Rd containing Zd, in this section we will present resultsproved in [ACP11] about characterizations of M−invariance of SIS V. We refer to thiswork for more details and proofs.
In order to give these characterizations, the authors construct a partition Bσσ∈N of Rd,where each Bσ will be an M∗−periodic set and the index set N will be properly chosenlater. Using this partition, for each σ ∈ N , they define the subspaces
Vσ = f ∈ L2(Rd) : f = χBσ g, whit g ∈ V. (3.3)
Using this partition, the authors characterize the M−invariance of V in terms of thesubspaces Vσ.
Theorem 3.3.1. If V ⊆ L2(Rd) is a SIS and M is a closed subgroup of Rd containing Zd,then the following are equivalent.
(i) V is M−invariant.
(ii) Vσ ⊆ V for all σ ∈ N .
Moreover, in case of any of the above holds, we have that V is the orthogonal direct sum
V =⊕
σ∈NVσ.
Remark 3.3.2. From the above theorem we deduce that a SIS V is M−invariant if and onlyif
V =⊕
σ∈NVσ.
40 Extra invariance of shift invariant spaces
The following lemma is used by the authors in [ACP11] to prove the above theorem.We will not give here the proof of this theorem but we will include the lemma because itresults interesting by itself.
Lemma 3.3.3. Let V be a SIS and σ ∈ N . Suppose that the subspace Vσ defined in (3.3)satisfies Vσ ⊆ V. Then, Vσ is a closed subspace which is M−invariant and in particular isa SIS.
Now we define the partition Bσσ∈N , in such a way that each Bσ is an M∗−periodic set.
Let Ω be a section of the quotient Rd/Zd given by
Ω = (T ∗)−1([0, 1)d),
where T is as in Remark 3.2.9. Then Ω tiles Rd by Zd translations, that is,
Rd =⋃k∈Zd
Ω + k.
We observe that, although the sets (Ω + k) + M∗k∈Zd are M∗−periodic, they are not apartition of Rd. So, we need to choose a subset N ⊆ Zd such that if σ,σ′ ∈ N andσ + M∗ = σ′ + M∗, then σ = σ′. That implies that N is a section of the quotient Zd/M∗.For example, we can define N by
N = (T ∗)−1(0, . . . , a1 − 1 × · · · × 0, . . . , aq − 1 × Zd−q),
where a1, . . . , aq are as in Therem 3.2.8. Then, given σ ∈ N , we define
Bσ := Ω + σ + M∗ =⋃
m∗∈M∗(Ω + σ) + m∗. (3.4)
Example 3.3.4. Let M = 1nZ ⊆ R. Then M∗ = nZ, Ω = [0, 1) and N = 0, . . . , n − 1.
Therefore,
Bσ =⋃
m∗∈nZ
([0, 1) + σ) + m∗ =⋃j∈Z
[σ,σ + 1) + n j, for each σ ∈ 0, . . . , n − 1.
The following figure illustrates the partition for n = 4. In this, the black dots representthe set N and the set B2 is the one which appears in gray.
−7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8
Ω
3.3 Characterization of extra invariance of SIS in L2(Rd) 41
Example 3.3.5. Let M = 12Z × R. Then M∗ = 2Z × 0, Ω = [0, 1)2 and N = 0, 1 × Z.
Then,B(i, j) =
⋃k∈Z
([0, 1)2 + (i, j)) + (2k, 0), for each (i, j) ∈ N .
In the following figure we have the sets B(0,0), B(1,1) and B(1,−1) which are represented bythe squares painted in light gray, gray and dark gray respectively. The setN is representedby the black dots.
−7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7−2
−1
0
1
2
3
Ω
On the other hand, in [ACP11], the authors give a characterization of the M−invariancefor a finitely shift invariant spaces in terms of the fibers and also in terms of the Gramian.
For f ∈ L2(Rd) and σ ∈ N , we will denote f σ as the function defined by
f σ = fχBσ .
Let Pσ be the orthogonal projection onto S σ, where
S σ = f ∈ L2(Rd) : supp( f ) ⊆ Bσ.
Then, we have that
f σ = Pσ f and Vσ = Pσ(V) = f σ : f ∈ V.
Let V be a SIS and Vσ be defined as in (3.3). If Φ is a countable subset of L2(Rd) suchthat V = S (Φ), then for ω ∈ Ω, we define the subspace
span τϕσ(ω) : ϕ ∈ Φ. (3.5)
We observe that in case of Vσ is closed, it is a SIS and this subspace is the fiber spaceJVσ(ω) defined in Proposition 1.4.3.
Remark 3.3.6. We observe that τϕσ(ω) = χBσ(ω+k) ϕ(ω+k)k∈Zd . Then, since χBσ(ω+k) ,0 if and only if k ∈ σ + M∗,
χBσ(ω + k) ϕ(ω + k) =
ϕ(ω + k) if k ∈ σ + M∗
0 otherwise.
Therefore, if σ , σ′, JVσ(ω) is orthogonal to JVσ′ (ω) for a.e. ω ∈ Ω.
42 Extra invariance of shift invariant spaces
In [ACP11], the authors prove the following result that characterizes the M−invarianceof a SIS in terms of fibers.
Theorem 3.3.7. Let V be a SIS generated by a countable set Φ ⊆ L2(Rd). The followingstatements are equivalent.
(i) V is M−invariant.
(ii) τϕσ(ω) ∈ JV(ω) a.e. ω ∈ Ω for all ϕ ∈ Φ and σ ∈ N .
The authors also obtain a slightly simpler characterization of M−invariance for thefinitely generated case, in terms of the Gramian and the dimension function.
Theorem 3.3.8. If V is a finitely generated SIS generated by Φ, then the following state-ments are equivalent.
(i) V is M−invariant.
(ii) For almost every ω ∈ Ω, dimV(ω) =∑σ∈N dimVσ(ω).
(iii) For almost every ω ∈ Ω, rank(GΦ(ω)) =∑σ∈N rank(GΦσ(ω)),
where Φσ = ϕσ : ϕ ∈ Φ.
Capıtulo 4:Aproximacion de datos con subespacios con
extra invariancia(Resumen)
Sea H un espacio de Hilbert, S una clase de subespacios cerrados de H y F = f1, . . . , fm
un conjunto finito de elementos en H.
En este capıtulo estudiamos la existencia y mostramos como construir un subespaciooptimo S en la clase S que maximice la distancia a un conjunto de datos dado F , enel sentido que S minimiza el funcional E(F , S ) en S. El funcional esta definido de lasiguiente manera
E(F , S ) =
m∑j=1
‖ f j − PS f j‖2, (3.6)
donde PS denota la proyeccion ortogonal sobre el subespacio S .
La motivacion para buscar un subespacio optimo en S es que, en muchas situacionesse quiere modelar un cierto conjunto de datos. En lugar de imponer ciertas condicionessobre los datos para que estos se adapten a un cierto modelo conocido, la idea es definiruna clase amplia conveniente de subespacios y encontrar un elemento de esa clase que “mejor aproxime ” al conjunto de datos.
Las senales a ser modeladas forman tipicamente un conjunto de dimension pequenapero dentro de un espacio de dimension mas grande. Sin embargo, como en las aplica-ciones las senales son observadas con ruido, se convierten en un conjunto de dimensiongrande pero cercano a un subespacio de dimension pequena que es el que se busca.
Cuando el espacio de Hilbert que se considera es L2(Rd) resulta natural considerar comomodelo para nuestro conjunto de datos la clase de los espacios invariantes por traslacionesenteras. Estos espacios se han usado en teorıa de aproximacion, analisis armonico, teorıade wavelets, teorıa de muestreo y procesamiento de imagenes (ver, e.g., [AG01, Gro01,HW96, Mal89] y las referencias que en ellos se encuentran).
En las aplicaciones a menudo se supone que las senales a estudiar pertenecen a ciertoespacio invariante por traslaciones enteras V generado por las traslaciones enteras de un
44 Resumen del Capıtulo 4
conjunto finito de funciones Φ = ϕ1, . . . , ϕm, i.e., V = S (Φ) = spanTkϕi : k ∈ Zd, i =
1, . . . ,m.
La eleccion del espacio invariante por traslaciones enteras finitamente generado partic-ular tipicamente no se deduce del conjunto de senales dado. Por ejemplo, en teorıa demuestreo, una suposicion clasica es que el conjunto de senales a ser muestreadas sean debanda limitada, es decir, las mismas pertenezcan a un espacio invariante por traslacionesenteras V generado por ϕ(x) = sinc(x). Sin embargo, la suposicion de ser de banda limi-tada no es muy aplicable en muchas aplicaciones. Luego, resulta natural buscar espaciosinvariantes por traslaciones enteras finitamente generados mas cercanos a un conjunto dedatos dado.
En este capıtulo estudiaremos este problema para el caso donde H = L2(Rd). Para estecaso, consideraremos como clase de subespacios aproximantes, la clase de los espaciosinvariantes por traslaciones enteras con extra-invariancia, es decir: si M es un subgrupode Rd que contiene a Zd, consideraremos subespacios S (ϕ1, . . . , ϕm) que sean M extra-invariantes. Luego, el espacio S (ϕ1, . . . , ϕm) resulta invariante no solo bajo traslacionesenteras, a pesar de estar generado por las traslaciones enteras de un conjunto finito defunciones.
Primero vamos a considerar el caso donde el subgrupo M esta contenido estrictamenteen Rd y contiene a Zd. Para este caso probamos que para cualquier conjunto finito dedatos F = f1, . . . , fm ⊆ L2(Rd), para cualquier subgrupo propio M que contiene a Zd
y para cualquier ` ∈ N, siempre existe un espacio invariante por traslaciones enteras Vde longitud a lo sumo ` con extra-invariancia M cuya distancia (en el sentido de (3.6))al conjunto de datos F es lo mas chica posible, si minimizamos sobre todos los espaciosinvariantes por traslaciones enteras de longitud menor o igual que ` que son M extra-invariantes.
Mostramos como construir una solucion V para este problema de aproximacion y ex-hibimos para este espacio solucion un conjunto de generadores cuyas traslaciones enterasforman un marco ajustado de V. Ademas damos una expresion para el error E(F ,V) entreel conjunto de datos y el espacio optimo, en funcion de los autovalores de una matrizespecial.
Si en vez de tomar M como un subgrupo propio de Rd, consideramos el caso dondeM = Rd, estamos en el caso en que la extra-invariancia es total. Es decir, espacios gener-ados por las traslaciones enteras de un numero finito de funciones que son Rd−invariantes.En este caso, un teorema de Wiener afirma que el espacio resulta isometrico a L2(Ω) paraalgun subconjunto medible Ω ⊆ Rd. Estos son espacios de Paley-Wiener generalizados.Para estos espacios probamos que, si las traslaciones enteras de los generadores formanuna base de Riesz, entonces el subconjunto Ω asociado es un “multi-tile” de Rd. Estudi-amos nuestro problema de aproximacion para estos espacios de Paley-Wiener generaliza-dos. Exhibimos para este caso como construir un conjunto de generadores del espaciooptimo y ademas mostramos una conexion interesante con resultados recientes acerca debases de exponenciales.
Resumen del Capıtulo 4 45
Finalmente consideramos un problema similar a los dos anteriores pero para el casodonde el espacio de Hilbert es `2(Zd) y la S es una clase de subespacios elegida con-venientemente. Obtenemos para este caso resultados equivalentes a los hallados para elcaso de L2(Rd). Resulta importante observar que el problema de aproximacion para elcaso discreto esta relacionado con el caso continuo como describiremos en Subsection4.3.3.
El problema de aproximar un conjunto de datos dado por espacios invariantes por trasla-ciones enteras (sin imponer ninguna restriccion acerca de la extra-invariancia) comenzoen [ACHM07] donde los autores probaron la existencia de un subespacio minimizantepara (3.6) sobre la clase de subespacios de dimension pequena en un espacio de HilbertH y ademas sobre la clase de los espacios invariantes por traslaciones enteras en L2(Rd).
En [ACM08] se considera el caso de multiples subespacios para el caso finito dimen-sional. Es decir, los autores construyen una union de subespacios de dimension pequenaque mejor aproxima a un conjunto de datos dado en Rd y ademas presentan un algoritmopara hallar dicha union. En el 2011, usando tecnicas de reduccion dimensional, este algo-ritmo es mejorado (ver [AACM11]).
Ademas, en [AT11] los autores encuentran condiciones necesarias y suficientes para laexistencia de subespacios optimos en el contexto general de espacios de Hilbert. Sin em-bargo, los autores no exhiben como puede llevarse a cabo la construccion de los mismos.
El primer resultado sobre aproximacion de un conjunto finito de datos usando espaciosinvariantes por traslaciones enteras con extra-invariancia aparece en [AKTW12], dondelos autores consideran espacios invariantes por traslaciones enteras principales en unavariable y suponiendo ademas que los espacios poseen un generador con traslacionesenteras ortogonales.
El capıtulo esta organizado de la siguiente forma. En Section 4.2 repasamos algunosresultados previos referentes al problema de encontrar un subespacio mas cercano a unconjunto de datos dado. Ademas, incluimos otros resultados que necesitaremos para laspruebas. Los principales resultados del capıtulo son enunciados y probados en Section4.3. En Subsection 4.3.1 presentamos el caso de espacios invariantes por traslacionesenteras que son M extra-invariantes, en Subsection 4.3.2 consideramos el caso de espaciosde Paley-Wiener generalizados y en Subsection 4.3.3 estudiamos el caso discreto.
4
Subspaces with extra invariance nearest toobserved data
4.1 Introduction
Let H be a Hilbert space, S a class of closed subspaces of H and F = f1, . . . , fm a finiteset of elements in H.
In this chapter we study the existence and show how to construct an optimal subspaceS in the class S that minimizes the distance to the given data F , in the sense that Sminimizes the functional E(F , S ) over S. The functional is defined as
E(F , S ) :=m∑
j=1
‖ f j − PS f j‖2, (4.1)
where PS denotes the orthogonal projection on the subspace S .
The motivation to find an optimal subspace in S is, that in many situations one wantsto choose a model for a certain class of data. Instead of imposing some conditions on thedata to fit some known model, the idea is to define a large class of subspaces convenientfor the application at hand, and find from there the one that “best fits” the data understudy.
The signals that need to be modelled are ideally low dimensional but living in a highdimensional space. However, since in applications they are often corrupted by noise, theybecome high dimensional, however they are close to a low dimensional subspace, whichis the space one seeks.
When the Hilbert space is L2(Rd) it is natural to consider as a model for our datathe class of shift invariant spaces. These spaces have been used in approximation the-ory, harmonic analysis, wavelet theory, sampling theory and signal processing (see, e.g.,[AG01, Gro01, HW96, Mal89] and references therein). Often, in applications, it is as-sumed that the signals under study belong to some shift invariant space V generated by thetranslations of a finite set of functions Φ = ϕ1, . . . , ϕm, i.e., V = S (Φ) = spanTkϕi : k ∈Zd, i = 1, . . . ,m.
48 Subspaces with extra invariance nearest to observed data
The choice of the particular finitely generated shift invariant space typically is not de-duced from a set of signals. For example in sampling theory, a classical assumption isthat the signals to be sampled are band-limited, that is, they belong to the shift invariantspace V generated by ϕ(x) = sinc(x). However, the band-limited assumption is not veryrealistic in many applications. Thus, it is natural to search for a finitely generated shiftinvariant space that is nearest to a set of some observed data.
In this chapter we study the case when H = L2(Rd). For this case, we restrict the classof approximating subspaces to be shift invariant spaces that have extra-invariance, that is:If M is a subgroup of Rd such that Zd ⊆ M we consider subspaces S (ϕ1, . . . , ϕm) M extra-invariant. Therefore, the space S (ϕ1, . . . , ϕm) is invariant under translates other than theintegers, even though it is generated by the integer translates of a finite set of functions.Such spaces with extra-invariance are important in applications specially in those wherethe jitter error is an issue.
We first consider the case when the subgroup M is a proper subgroup ofRd that containsZd. For that case we obtain one of the main contribution of this chapter. We prove that forany finite set of data F = f1, . . . , fm ⊆ L2(Rd), for any proper subgroup M containing Zd
and for any ` ∈ N there always exists a SIS V of length at most ` with extra invariance Mwhose distance (in the sense of (4.1)) to the data F is the smallest possible among all theSIS of length smaller or equal than ` that are M extra-invariant.
We construct a solution V and provide a set of generators whose integer translates forma tight frame of V. An expression for the exact value of the error E(F ,V) between the dataand the optimal subspace is also obtained using the eigenvalues of some special matrix.
If instead of M being a proper subgroup of Rd we consider M = Rd, we are in the caseof total extra-invariance. That is, spaces generated by integer translates of a finite numberof functions that are Rd invariant. In this case, a theorem of Wiener says that the spaceis isometric to L2(Ω) for some measurable subset Ω ⊆ Rd. These are generalized Paley-Wiener spaces. We prove that for these spaces, if the integer translates of the generatorsform a Riesz basis then the associated set Ω multi-tilesRd by translates along Zd.We studyour approximation problem for those generalized Paley-Wiener spaces. We describe forthis case how to construct a set of generators and show an interesting connection withrecent results about bases of exponentials.
Finally we consider a similar problem when the Hilbert space is `2(Zd) and S is aconveniently chosen class of subspaces. We obtain for this case equivalent results tothe ones for L2(Rd). The approximation problem for the discrete case is related with thecontinuous case in a very interesting way that is described in Subsection 4.3.3.
The problem of approximation of a set of data by shift invariant spaces (without theextra invariance restriction) started in [ACHM07] where the authors proved the existenceof a minimizer for (4.1) over the class of low dimensional subspaces in a Hilbert space Hand also over the class of shift invariant spaces in L2(Rd).
In [ACM08] the case of multiple subspaces was considered in the finitely dimensional
4.2 Preliminaries 49
case. That is, the authors found a union of low dimensional subspaces that best fits agiven set of data in Rd and provided an algorithm to find it. In 2011, using dimensionalreduction techniques, this algorithm was improved (see [AACM11]).
Further, in [AT11] the authors found necessary and sufficient conditions for the exis-tence of optimal subspaces in the general context of Hilbert spaces. However, they didnot provide a way to construct them.
The first result for approximation of a finite set of data using shift invariant spaces withextra-invariance constrains appears in [AKTW12], where the authors consider principalshift invariant spaces in one variable and they assume that the space has a generator withorthogonal integer translates.
The rest of this chapter is organized as follows. In Section 4.2 first we review someprevious work about finding a subspace that is closest to a given data set. Also, we setsome other results that we need. The main results of this chapter are stated and proved inSection 4.3. In Subsection 4.3.1 we present the M extra-invariant case for shift invariantspaces, in Subsection 4.3.2 we consider the case of Paley-Wiener spaces and in Subsection4.3.3 we study a discrete case.
4.2 Preliminaries
We begin with a review about previous related work. The results will be establishedwithout proofs, but we will provide references where the proofs can be found.
Given a Hilbert space H, F = f1, . . . , fm a set of vectors in H, and a class of closedsubspaces S of H, the problem of finding a subspace S ∗ ∈ S that best approximates thedata F has several applications in mathematics, engineering and computer science.
Definition 4.2.1. We will say that a set of closed subspaces S of a separable Hilbertspace H has the Minimum Subspace Approximation Property (MSAP) if for every subsetF ⊆ H there exists an element S ∗ ∈ S such that minimizes the expression
E(F , S ) =∑f∈S
d2( f , S ), over all S ∈ S. (4.2)
Any subspace S ∗ ∈ S satisfying (4.2) will be called an optimal subspace for F .
In [AT11], the authors provide necessary and sufficient conditions on the class S ofclosed subspaces in a separable Hilbert space H so that S satisfies the MSAP.
First we begin studying the problem of finding optimal subspaces in the case H = Rd
and S being the class of all subspaces of dimension at most ` (with ` << d). Then,we want to find a subspace S ∗ of dimension at most ` that is close to a given finte dataF = f1, . . . , fm ⊆ R
d in the sense that S ∗ satisfiesm∑
j=1
‖ f j − PS ∗ f j‖2 ≤
m∑j=1
‖ f j − PS f j‖2, ∀ S ∈ S`,
50 Subspaces with extra invariance nearest to observed data
where S` is the class of all subspaces of dimension at most `.
This problem is solved by Eckart-Young’s Theorem (see [EY36, Sch07]) which usesthe Singular Value Decomposition (SVD) of a matrix and its relation to finite dimensionalleast squares problems. Before stating the theorem, we will review the construction of theSVD of a matrix (for a detailed treatment see [Bha97, HJ85]).
Let A = [ f1, . . . , fm] ∈ Rd×m and r = rank(A). Consider the matrix A∗A ∈ Rm×m. SinceA∗A is self-adjoint and positive semi-definite, its eigenvalues λ1 ≥ · · · ≥ λm are nonnega-tive and the associated eigenvectors y1, . . . , ym can be chosen to form an orthonormal basisof Rm. We observe that the rank r of A corresponds to the largest index i such that λi > 0,then we have that λ1 ≥ · · · ≥ λr > 0 = λr+1 = · · · = λm. The left singular vectors u1, . . . , ur
can then be obtained from√λiui = Ayi, that is ui = λ−1/2
i
m∑j=1
yi jai j, 1 ≤ i ≤ r.
Here yi = (yi1, . . . , yim)t. The remaining left singular vectors ur+1, . . . , um can be cho-sen to be any orthonormal collection of m − r vectors in Rd that are perpendicular tospan f1, . . . , fm. Then we can obtain the following SVD of A
A = UΛ1/2Y∗,
where U ∈ Rd×m is the matrix with columns u1, . . . , um, Λ1/2 = diag (λ1/21 , . . . , λ1/2
m ) andY ∈ Rm×m is the matrix with columns y1, . . . , ym, with U∗U = Im = YY∗.
Then, we state the Eckart-Young’s Theorem.
Theorem 4.2.2. Let F = f1, . . . , fm be a set of vectors in Rd, A = [ f1, . . . , fm] ∈ Rd×m,with r = rank(A). Suppose that A has SVD A = UΛ1/2Y∗ and that 0 < ` ≤ r. If S ∗ =
span u1, . . . , u`, thenm∑
j=1
‖ f j − PS ∗ f j‖2 ≤
m∑j=1
‖ f j − PS f j‖2, ∀ S ∈ S`.
Moreover,
E(F ,S`) =
r∑j=`+1
λ j,
where λ1 ≥ · · · ≥ λr > 0 are the positive eigenvalues of AA∗.
Then, we have that in the Hilbert space H = Rd, the class S` of all subspaces ofdimension at most ` has the MSAP. Therefore, given a finite data F = f1, . . . , fm ⊆ R
d,there exists S ∗ ∈ S` such that approximates the data F . Also, the above theorem showshow to construct the generators of the optimal space S ∗ and gives an expression for theerror in this approximation.
In [ACHM07], the authors consider the above problem but in a general context. Moreprecisely, given a finite data F in an arbitrary Hilbert space H, they proved that therealways exists a subspace S ∗ with dimension at most ` that approximates the data F . Theystate the following theorem which is an extension of the Eckart-Young’s theorem.
4.2 Preliminaries 51
Theorem 4.2.3 (Theorem 4.1, [ACHM07]). Let H be a Hilbert space, F = f1, . . . , fm ⊆
H, and ` ∈ N. Define X = span f1, . . . , fm, and let λ1 ≥ · · · ≥ λm be the eigenval-ues of the matrix B(F ) defined as [B(F )]i, j = 〈 fi, f j〉H and y1, . . . , ym ∈ C
d, with yi =
(yi1 , . . . , yim)t orthonormal left eigenvectors associated to the eigenvalues λ1 ≥ · · · ≥ λm.Let r = dim X = rank B(F ). Define the vectors q1, . . . , q` ∈ H by
qi = θi
m∑j=1
yi j f j, i = 1, . . . , `
where θi = λ−1/2i if λi , 0 and θi = 0 otherwise. Then q1, . . . , q` is a Parseval frame of
S ∗ = span q1, . . . , q` and the subspace S ∗ is optimal in the sense that
m∑i=1
‖ fi − PS ∗ fi‖2 ≤
m∑i=1
‖ fi − PS fi‖2, ∀ S ∈ S`.
Furthermore we have the following formula for the error
E(F ,S`) =
m∑i=`+1
λi.
We observe that if r is small, that is, r ≤ `, then all the vectors qr+1, . . . , q` are null andq1, . . . , qr is an orthonormal set.
Moreover, one could choose qr+1, . . . , q` to be any orthonormal set in the orthogonalcomplement of X and so obtain an orthonormal set of ` elements and the formula for theerror in the above theorem would still hold.
Then, using Theorem 4.2.3, we obtain that the class S` of all subspaces of dimensionat most ` has the MSAP in an arbitrary Hilbert space H. Also, the authors give a way toconstruct the generators of an optimal subspace and present a formula for the error in thisapproximation.
Now, we consider the problem of finding optimal subspaces in the case H = L2(Rd)and S being the class of all shift invariant spaces of length at most `. We will denotethis class V`. Then, we want to find a SIS V∗ ∈ V` that is closed to a given finite dataF = f1, . . . , fm ⊆ L2(Rd), in the sense that V∗ satisfies
m∑j=1
‖ f j − PV∗ f j‖2 ≤
m∑j=1
‖ f j − PV f j‖2, ∀V ∈ V`, (4.3)
and estimate the error of this approximation.
Observe that in the two above cases, we take the minimum over all subspaces of di-mension at most `, while in this problem the minimization is taken over a particular classof infinite dimensional subspaces, so the problems are essentially different, so the SVDcannot be applied directly. However, due to the special structure of shift invariant spaces,
52 Subspaces with extra invariance nearest to observed data
the Fourier transform converts this problem into finite dimensional least square problemsat each frequency. The following theorem shows that the above problem has a solution,that is, there exists a SIS of length at most ` which satisfies (4.3). Recall from Definition1.5.2 that for a given set F = f1, . . . , fm ⊆ L2(Rd), the Gramian GF ∈ Cm×m is defined as[GF (ω)]i j = 〈τ f j(ω), τ fi(ω)〉`2(Zd) for all 1 ≤ i, j ≤ m, where τ f (ω) = f (ω + k)k∈Zd .
Theorem 4.2.4 (Theorem 2.3, [ACHM07]). Let F = f1, . . . , fm be a set of functions inL2(Rd). Let λ1(ω) ≥ · · · ≥ λm(ω) be the eigenvalues of the Gramian GF (ω). Then, thereexists V∗ ∈ V` such that
m∑i=1
‖ fi − PV∗ fi‖2 ≤
m∑i=1
‖ fi − PV fi‖2, ∀V ∈ V`.
Moreover, we have that
(1) The eigenvalues λi(ω), 1 ≤ i ≤ m are Zd−periodic, measurable functions inL2([0, 1]d) and
E(F , `) =
m∑i=`+1
∫[0,1]d
λi(ω) dω.
(2) Let θi(ω) = λ−1i (ω) if λi(ω) is different from zero, and zero otherwise. Then,
there exists a choice of measurable left eigenvectors Y1(ω), . . . ,Y`(ω) with Y i =
(yi1, . . . , y
im)t, i = 1, . . . , `, associated with the first ` largest eigenvalues of GF (ω)
such that the functions defined by
ϕi(ω) = θi(ω)m∑
j=1
yij(ω) f j(ω), i = 1, . . . , `, ω ∈ Rd
are in L2(Rd). Furthermore, the corresponding set of functions Φ = ϕ1, . . . , ϕ` is agenerator set for the optimal subspace V∗ and the set ϕi(· − k), k ∈ Zd, i = 1, . . . , `is a Parseval frame for V∗.
Then, we have that the classV` of finitely generated SIS of L2(Rd) with length at most` has the MSAP.
As we mention above, in order to find a solution of this new problem, the authors reduceit into an uncountable set of finite dimensional problems in the Hilbert space H = `2(Zd),which can be solved using Theorem 4.2.3. Finally, they construct the generators of theoptimal space patching together the solutions of the reduced problems to obtain a solutionto the original problem.
For the proof they also use the following technical lemma concerning the measurabilityof the eigenvalues and the existence of measurable eigenvectors of a non-negative matrixwith measurable entries. We will include here this result since we will need it later forprove our results.
4.2 Preliminaries 53
Lemma 4.2.5 (Lemma 2.3.5, [RS95]). Let G(ω) be an m × m self-adjoint matrix of mea-surable functions defined on a measurable subset E ⊆ Rd with eigenvalues λ1(ω) ≥ · · · ≥λm(ω). Then the eigenvalues λi, i = 1, . . . ,m, are measurable functions on E and thereexists an m × m matrix of measurable functions U(ω) on E such that U(ω)U∗(ω) = I a.e.ω ∈ E and such that
G(ω) = U(ω)Λ(ω)U∗(ω), a.e. ω ∈ E,
where Λ(ω) := diag(λ1(ω), . . . , λm(ω)).
Now, we consider the problem of finding a union of low dimensional subspaces thatbest fits a given set of data in a Hilbert space H.
Since we are interested in models that are union of subspaces, we will arrange thesubspaces in finite bundles that will be our main objects, and define the error between abundle and a set of vectors.
Let us fix m, n ∈ N with 1 ≤ n ≤ m, let F = f1, . . . , fm ⊆ H and let S be a family ofclosed subspaces of H containing the zero subspace. Define the collection of bundles Bto be the set of sequences of elements in S of length n, that is,
B = V = V1, . . . ,Vn : Vi ∈ S, 1 ≤ i ≤ n.
For V ∈ B, with V = V1, . . . ,Vn, we define the error between a bundle V and F asfollows
e(F ,V) =
m∑j=1
min1≤i≤n
d2( f j,Vi),
where d is the distance in H. Note that e(F ,V) is a non-linear function of F . Also, weobserve that, in the case n = 1, this error agrees with the error given in the above problems,that is,
E(F , V) = e(F ,V) =
m∑j=1
d2( f j,V).
The following theorem shows that the problem of finding an optimal union of subspaceshas solution for every finite set of data F ⊆ H and every n ≥ 1 if and only if the class Shas the MSAP.
Theorem 4.2.6 ([ACM08]). Let F = f1, . . . , fm be a vectors in H and let n be given(with n < m). If S satisfies the MSAP, then there exists a bundle V∗ = V∗1 , . . . ,V
∗n ∈ B
such thate(F ,V∗) = infe(F ,V) : V ∈ B. (4.4)
Any V∗ ∈ B that satisfies (4.4) will be called an optimal bundle for F .
As we mentioned above, the class V` of finitely generated shift invariant spaces ofH = L2(Rd) with length at most ` has the MSAP. Then, by Theorem 4.2.6, there exists asolution for the problem of optimal union of finitely generated SISs.
54 Subspaces with extra invariance nearest to observed data
On the other hand, by Theorem 4.2.3, we obtain that the class S` of all subspaces ofdimension at most ` has the MSAP in an arbitrary Hilbert space H. Then using Theorem4.2.6 we have that there exists a union of `−dimensional subspaces which is closest to agiven data set.
Note that e(F ,V) in Theorem 4.2.6 is computed as follows: For each f ∈ F find thespace V j( f ) in V closest to f , compute d2( f ,V j( f )), and the sum over all values found byletting f run throughF . This procedure is not feasible in practice and a search algorithm isneeded. So, in [ACM08], the authors provided an algorithm to find the solution. In 2011,using dimension reduction techniques, this algorithm was improved (see [AACM11]). Wewill not include here all the results concerning these algorithms since we would need alot of notation and technical aspects which are not congruent with the general line of thisthesis. For a complete treatment, we refer the reader to [AACM11, ACM08].
Finally, we consider optimal models with extra invariance. We observe that, althoughTheorem 4.2.4 shows the existence of a finitely generated shift invariant space which isclosest to a finite set of data, it does not provide an answer if we require the shift invariantspace to have some extra invariance. In this chapter we prove that given an arbitraryfinite set of data it always exists a finitely shift invariant space with extra invariance thatminimize the distance to the data. Also we will consider a similar problem for the case oftotal extra invariance.
The first result about approximation using shift invariant spaces with extra invarianceconstrains appear in [AKTW12], where the authors consider principal shift invariantspaces in L2(R) and they suppose that the space has an orthonormal generator ϕ, thatis, ϕ(· − k) : k ∈ Z is an orthonormal basis for the space S (ϕ).
As we mentioned in Chapter 3, principal shift invariant spaces can also be invariantunder a set of translates that strictly contains Z. Also, we know that, in the real line, theinvariance set MV for a shift invariant space V is R or 1
nZ for some n ∈ N. In [AKTW12],the authors denote byV the class of all principal SIS with an orthonormal generator whichare also translation invariant (that is, MV = R). Also, they considerVn being the class ofall principal SIS with an orthonormal generator which are also 1
nZ−invariant. With thisnotations they prove the following result.
Theorem 4.2.7. Let n ≥ 2 and let F = f1, . . . , fm ⊆ L2(R) be a set of functions. Then
(i) There exists V∗ = S (ϕ) ∈ V such that
m∑j=1
‖ f j − PV∗ f j‖2 ≤
m∑j=1
‖ f j − PV f j‖2, for all V ∈ V.
(ii) There exists V∗ = S (ϕ) ∈ Vn such that
m∑j=1
‖ f j − PV∗ f j‖2 ≤
m∑j=1
‖ f j − PV f j‖2, for all V ∈ Vn.
4.3 Main results 55
We want to remark here that a key element in their proof is that the space is principaland the generator has orthogonal integer translates. So theirs techniques do not apply toour general case.
4.3 Main results
4.3.1 Optimality for the class of SIS with extra-invariance
In this subsection we consider the approximation problem for the class of finitely gener-ated SIS with extra invariance under a given proper subgroup M of Rd.
Let us start introducing some notation. Let m, ` ∈ N, M be a closed proper subgroup ofRd containing Zd and F = f1, . . . , fm ⊆ L2(Rd). Define
V`M := V : V is a SIS of length at most ` and V is M-invariant. (4.5)
LetN = σ1, . . . , σκ be a section of the quotient Zd/M∗ and Bσ : σ ∈ N the partitiondefined in (3.4).
For each σ ∈ N , we consider F σ = f σ1 , . . . , f σm ⊆ L2(Rd) where, f σj is such that
f σj = f jχBσ for j = 1, . . . ,m. Also, let
F = f σ11 , . . . , f σ1
m , . . . . . . , f σκ1 , . . . , f σκm .
For each ω ∈ U ⊆ Rd (a measurable set of representatives of the quotient Rd/Zd) letGF
(ω) be the associated Gramian matrix of the vectors in F with eigenvalues
λ1(ω) ≥ · · · ≥ λmκ(ω) ≥ 0.
Using Lemma 4.2.5, we have that these eigenvalues are measurable functions.
Since f σsi is orthogonal to f σt
i if s , t, the Gramian GF
(ω) is a diagonal block matrixwith blocks Gσ(ω), σ ∈ N . Here Gσ(ω) denotes the m × m Gramian associated to thedata F σ.
On the other hand, using Lemma 4.2.5 again, we have that
Gσ(ω) = Uσ(ω)Λσ(ω)U∗σ(ω) a.e. ω ∈ U
where Uσ are unitary and Λσ(ω) := diag(λσ1 (ω), . . . , λσm(ω)) ∈ Cm×m and they are alsomeasurable matrices as in Lemma 4.2.5. We also have λσ1 (ω) ≥ · · · ≥ λσm(ω) for eachσ ∈ N .
Using the decompositions of the blocks Gσ we have that
GF
(ω) = U(ω)Λ(ω)U∗(ω) (4.6)
where U has blocks Uσ in the diagonal, and Λ is diagonal with blocks Λσ.
56 Subspaces with extra invariance nearest to observed data
We want to recall here that for almost each ω ∈ U the matrix Λ(ω) collects all theeigenvalues of the Gramian G
F(ω) and the columns of the matrix U(ω) are the associated
left eigenvectors. Note that an eigenvector associated to the eigenvalue λσj (ω) has all thecomponents not corresponding to the block σ equal to zero.
Now for each fixed ω ∈ U, we consider (i1(ω), j1(ω)), . . . , (in(ω), jn(ω)) with is(ω) ∈N and js(ω) ∈ 1, . . . ,m and n = mκ such that
λi1(ω)j1(ω) ≥ · · · ≥ λ
in(ω)jn(ω) ≥ 0
are the ordered eigenvalues of GF
(ω), with corresponding left eigenvectors Y (is(ω), js(ω)) ∈
Cn, for s = 1, · · · , n.
Here is(ω) indicates the block of the matrix GF
(ω) in which the eigenvalue λis(ω)js(ω)(ω)
is found and js(ω) indicates the displacement in this block of the matrix GF
(ω). Moreprecisely, we have that λis(ω)
js(ω)(ω) coincides with λ(is(ω)−1)m+ js(ω)(ω), the ((is(ω) − 1)m +
js(ω))−th eigenvalue of GF
(ω). When ω ∈ U is fixed, we will write is instead of is(ω)and js instead of js(ω).
We will prove now that γs(ω) := λis(ω)js(ω)
(ω) is measurable as a function on ω for eachs = 1, . . . , n.
Let s ∈ 1, . . . , n fixed. Let is(ω) ∈ N and js(ω) ∈ 1, . . . ,m. We have that γs(ω) =
λσj (ω) for all ω ∈ Eσ j := ω ∈ U : is(ω) = σ, js(ω) = j.
We observe that
Eσ j = ω ∈ U : is(ω) = σ, js(ω) = j = ω ∈ U : λs(ω) = λσj (ω).
Using Lemma 4.2.5 applied to GF
(ω) and Gσ(ω), we have that λs and λσj are measurablefunctions of ω. Therefore Eσ j are measurable sets for all σ ∈ N and for all j ∈ 1, . . . ,m.
We further observe that γs(ω) = λσj (ω), for ω ∈ Eσ j. So γs(ω) is a measurable function.A similar argument shows that the eigenvectors are measurable.
Finally we define hs : Rd → C, for s = 1, . . . , `
hs(ω) := θisjs(ω)
m∑k=1
y(is, js)(is−1)m+k(ω) f is
k (ω), (4.7)
where θisjs(ω) = (λis
js(ω))−1/2 if λis
js(ω) , 0 and θis
js(ω) = 0 otherwise.
Now we are ready to state the main result of this section.
Theorem 4.3.1. Let m, ` ∈ N, and M be a closed proper subgroup of Rd containing Zd.Assume that F = f1, . . . , fm ⊆ L2(Rd) is given data and let V`
M be the class defined in(4.5). Then, there exists a shift invariant space V∗ ∈ V`
M such that
V∗ = argminV∈V`
M
m∑j=1
‖ f j − PV f j‖2. (4.8)
Furthermore, with the above notation,
4.3 Main results 57
(1) The eigenvalues λσj (ω) : σ ∈ N , j = 1, . . . ,m, are Zd−periodic, measurablefunctions in L2(U) and the error of approximation is
E(F ,M, `) :=m∑
j=1
‖ f j − PV∗ f j‖2 =
∫U
mκ∑s=`+1
λisjs(ω) dω.
(2) The functions h1, . . . , h` defined in (4.7) are in L2(Rd) and if ϕ1, . . . , ϕ` are definedby ϕ j = h j, then Φ = ϕ1, · · · , ϕ` is a generator set for the optimal subspace V∗
and the set ϕi(· − k), k ∈ Zd, i = 1, . . . , ` is a Parseval frame for V∗.
Proof. Let V` be the class as in Theorem 4.2.4, that is V` is the set of all shift invariantspaces V that can be generated by ` or less generators. (Note that we do not ask theelements of the classV` to have extra invariance.)
Define V∗ ∈ V` to be the optimal space given by Theorem 4.2.4 for the data F . Thatis, V∗ satisfies∑
σ∈N
m∑j=1
‖ f σj − PV∗ f σj ‖2 ≤
∑σ∈N
m∑j=1
‖ f σj − PV f σj ‖2 ∀V ∈ V`. (4.9)
We claim that V∗ ∈ V`M (in particular V∗ is M extra-invariant) and it is optimal in this
class for the data F , i.e.m∑
j=1
‖ f j − PV∗ f j‖2 ≤
m∑j=1
‖ f j − PV f j‖2 ∀V ∈ V`
M. (4.10)
Let us prove first that V∗ is M extra-invariant. For this we will check that the generatorsof V∗ satisfy condition (ii) in Theorem 3.3.7. We have from (4.6) that the Gramian G
F(ω)
can be decomposed as GF
(ω) = U(ω)Λ(ω)U∗(ω) with eigenvalues λσj (ω) : σ ∈ N , j =
1, . . . ,m.
By Theorem 4.2.4, the ` generators of V∗ have the form defined in (4.7),
ϕs(ω) = θisjs(ω)
m∑k=1
y(is, js)(is−1)m+k(ω) f is
k (ω), for s = 1, . . . , `. (4.11)
From (4.11) it is clear that ϕs is supported in Bis , since each f isk is supported in Bis . Then
if we apply the cut off operator to these generators we obtain
ϕσs (ω) =
ϕs(ω) if σ = is(ω),0 otherwise.
The above implies that
τϕσs (ω) =
τϕs(ω) if σ = is(ω),0 otherwise.
58 Subspaces with extra invariance nearest to observed data
So, in any case τϕσs (ω) ∈ JV∗(ω) for a.e. ω ∈ U and for all σ ∈ N , s = 1, . . . , ` whichproves the M-invariance of V∗.
What is left now is to prove that V∗ is optimal over the class V`M, that is V∗ satisfies
equation (4.10). For this note that if V ∈ V`M then V =
⊕σ∈N
Vσ. So we have for anyf ∈ L2(Rd),
‖PV f ‖2 = ‖PV
∑σ∈N
f σ‖2 = ‖∑σ∈N
PV f σ‖2 = ‖∑σ∈N
PVσ f σ‖2 =∑σ∈N
‖PVσ f σ‖2 =∑σ∈N
‖PV f σ‖2,
which implies together with (4.9) that
m∑j=1
‖PV∗ f j‖2 ≥
m∑j=1
‖PV f j‖2, ∀V ∈ V`
M.
The others claims of the theorem are a direct consequence of Theorem 4.2.4.
4.3.2 Approximation with Paley-Wiener spaces
Preliminaries
In this section the class of approximation subspaces will be finitely generated SIS withtotal translation invariance. That is translation invariant spaces that are generated by theinteger translates of a finite number of functions.
More precisely, given ` ∈ N define T ` to be the set of all shift invariant spacesV = S (ϕ1, . . . , ϕ`) for some functions ϕ1, . . . , ϕ` in L2(Rd), and such that V is transla-tion invariant and the integer translates of ϕ1, . . . , ϕ` form a Riesz basis of V .
Given a set F = f1, . . . , fm ⊆ L2(Rd), we want to find V∗ ∈ T ` such that
V∗ = argminV∈T `
m∑j=1
‖ f j − PV f j‖2. (4.12)
Here PV denotes the orthogonal projection on V.
Before going to the approximation problem, we will obtain a characterization of theclass T `.
Using Wiener’s theorem, we have that V is a translation invariant space in L2(Rd) if andonly if there exists a measurable set Ω ⊆ Rd such that
V = f ∈ L2(Rd) : f (ω) = 0 a.e. ω ∈ Rd \Ω.
Since Ω is unique up to measure zero, we will write V = VΩ.
4.3 Main results 59
Definition 4.3.2. Let Ω ⊆ Rd be measurable and L ⊆ Rd be a countable set. We say thatΩ tiles Rd when translated by L at level ` ∈ N if∑
t∈L
χΩ(ω − t) = ` for a.e. ω ∈ Rd.
In case of L = Zd we will say that Ω is an ` multi-tile.
It is known (see for example [Ko15],) that Ω is an ` multi-tile of Rd, if and only if,up to measure zero, Ω is the union of ` measurable and disjoint 1 tile sets. i.e. Ω is aquasi-disjoint union of ` sets of representatives of Rd/Zd.
Lemma 4.3.3. A measurable set Ω ⊆ Rd, ` multi-tiles Rd if and only if
Ω = Ω1 ∪ · · · ∪Ω` ∪ N,
where N is a zero measure set, and the sets Ω j, 1 ≤ j ≤ ` are measurable, disjoint andeach of them tiles Rd by translations on Zd.
The following proposition characterizes the set Ω for the elements in T `.
Proposition 4.3.4. A subspace V is in T ` if and only if V = VΩ with Ω a measurable `multi-tile of Rd.
Proof. Assume first that V ∈ T `, so V = VΩ for some measurable Ω ⊆ Rd. Also, as aconsequence of Wiener’s theorem, for almost all ω ∈ U we have JV(ω) `2(Oω) with
Oω = k ∈ Zd : w + k ∈ Ω.
To see this, we note that JV(ω) ⊆ `2(Oω). For the other inclusion, fix ω ∈ U. Using that
Ω =⋃k∈Zd
Ek where Ek = (U + k) ∩Ω,
we have that k ∈ Oω, if and only if ω+ k ∈ Ek. Hence, if a ∈ `2(Oω) consider the function
Gω(ξ) :=∑k∈Oω
akχEk(ξ).
Since Gω is in L2(Ω), the function g defined by g = Gω is in V , and g(ω + k) = ak ifk ∈ Oω. Therefore, g ∈ V and a = τg(ω) ∈ JV(ω).
Now, since V = S (ϕ1, . . . , ϕ`), and the integer translates of ϕ1, . . . , ϕ` form a Rieszbasis of V, using Theorem 1.5.1 we obtain that τϕ1(ω), . . . , τϕ`(ω) form a Riesz basis ofJV(ω) with the same Riesz bounds for a.e. ω ∈ U. We conclude that dim(JV(ω)) = ` a. e.ω ∈ U.
Since V is translation invariant, by the observation above dim(JV(ω)) = #Ow. Then#Oω = ` for almost all ω ∈ U, which implies that Ω is an ` multi-tile. (Here #A denotethe cardinal of the set A).
60 Subspaces with extra invariance nearest to observed data
For the converse, assume that Ω is a measurable ` multi-tile of Rd. Define V = VΩ. So,V is translation invariant.
By Lemma 4.3.3 we have that Ω = Ω1 ∪ · · · ∪ Ω` up to a measure zero set, where eachΩ j is a set of representatives or Rd/Zd. We define ϕ j by its Fourier transform:
ϕ j = χΩ j , j = 1, . . . , `.
Since e2πiωk ϕ j : k ∈ Zd is an orthonormal basis of L2(Ω j), we have that e2πiωkϕ j : k ∈Zd, j = 1, . . . , ` is an orthonormal basis of L2(Ω), and so, tkϕ j : k ∈ Zd, j = 1, . . . , ` isan orthonormal basis of V , in particular a Riesz basis.
The approximation problem for Paley-Wiener Spaces
Now we come back to our approximation problem. In order to find an optimal subspacein the class T ` for a set of data F = f1, . . . , fm, it is enough to find the associated `multi-tile Ω in Rd.
It is not difficult to see that if we allow Ω to be any ` multi-tile the minimum in (4.12)may not exist. So we will restrict Ω to be inside a cube that could be arbitrarily large. Letus fix N ∈ N. Define
CN := [−(N + 1/2),N + 1/2]d,
M`N := Ω ⊆ CN : Ω is measurable and ` multi-tiles Rd andT `
N := V ∈ T ` : V = VΩ with Ω ∈ M`N.
With this notation we can state the main result of this section.
Theorem 4.3.5. Assume that m, ` ∈ N and a set F = f1, . . . , fm ⊆ L2(Rd), are given.Then for each N ≥ ` there exists a Paley-Wiener space V∗ ∈ T `
N that satisfies
V∗ = argminV∈T `
N
m∑j=1
‖ f j − PV f j‖2, (4.13)
where T `N is the class defined above.
Proof. First we observe that if a solution space V∗ exists then
V∗ = argminV∈T `
N
m∑j=1
‖ f j − PV f j‖2 = argmax
V∈T `N
m∑j=1
‖PV f j‖2, (4.14)
and using the definition of T `N , we have that
maxV∈T `
N
m∑j=1
‖PV f j‖2 = max
Ω∈M`N
m∑j=1
‖PVΩf j‖
2. (4.15)
4.3 Main results 61
So, we need to find Ω ∈ M`N that yields the maximum in (4.15).
Using Lemma 1.4.4 we see that for each Ω ∈ M`N ,
m∑j=1
‖PVΩf j‖
2 =
m∑j=1
‖PVΩf j‖
2 (4.16)
=
m∑j=1
∫U
‖PJVΩ(ω)(τ f j(ω))‖2
`2(Zd) dω
=
∫U
m∑j=1
‖PJVΩ(ω)(τ f j(ω))‖2
`2(Zd) dω.
Recall that PJVΩ(ω) denotes the orthogonal projection onto the closed subspace JVΩ
(ω)of `2(Zd).
Furthermore, if Ω ∈ M`N , we know from the proof of Proposition 4.3.4 that
dim(JVΩ(ω)) = ` for a.e. ω ∈ U. Note that JVΩ
(ω) agrees with the subspace of `2(Zd)of the sequences supported in Oω. Then there exists a unique set of ` integer vectorskΩ(ω) := kΩ
1 (ω), . . . , kΩ` (ω) ⊆ Zd such that
spanδkΩj (ω) : j = 1, . . . , ` = JVΩ
(ω) for a.e. ω ∈ U.
Here δ j denotes the canonical vector in `2(Zd). i.e. δ j(s) = 0 if s , j and 1 otherwise.Note that, since Ω ⊆ CN necessarely ‖kΩ
j (ω)‖∞ ≤ N, for each j and ω. Combining thisobservation with (4.16) we obtain,
m∑j=1
‖PVΩf j‖
2 =
∫U
m∑j=1
∑s=1
| f j(ω + kΩs (ω))|2 dω. (4.17)
So, now we need to maximize the left hand side in (4.17) over all the sets Ω ∈ M`N .
Note that given Ω ∈ M`N , for almost each ω ∈ U, the set Ω contains exactly ` elements
from the sequence ω + k, k ∈ Zd. Then we can pick for each ω ∈ U (up to a set of zeromeasure) ` translations k∗s(ω) such that
∑mj=1
∑`s=1 | f j(ω+k∗s(ω))|2 is maximum over all sets
of ` translations k = k1, . . . , k` ⊆ Zd, with ‖k j‖∞ ≤ N. The maximum exists since thefibers of f j are `2(Zd)-sequences and the number of translations considered is finite.
Call K the set of admisibles translations i.e. K := k = k1, . . . , k` ⊆ Zd : ‖k j‖∞ ≤ Nand for k ∈ K set
Hk(ω) :=m∑
j=1
∑s=1
| f j(ω + ks(ω))|2.
Our goal is to construct a set Ω such that the associated space VΩ is optimal. So theidea is to construct the optimal set Ω∗ considering for each ω ∈ U the optimal translationsω + k∗s(ω) : s = 1, . . . , `, and then taking the union over almost all ω ∈ U.
62 Subspaces with extra invariance nearest to observed data
For this we define for each k = k1, . . . , k` ∈ K the following subset ofU,
Ek := ω ∈ U : Hk(ω) ≥ Hr(ω), ∀ r = r1, . . . , r` ∈ K ,
i.e., Ek is the set of ω ∈ U for which the maximum is attained for k = k1, . . . , k`. Notethat Ek could be the empty set for some k = k1, . . . , k` and the sets Ek may not bedisjoint.
Finally we define our optimal set as,
Ω∗ :=⋃k∈K
⋃j=1
Ek + k j.
We will now prove that Ω∗ is measurable. First we note that Ek is a measurable set foreach k ∈ K since,
Ek =⋂r∈K
Fkr ,
where,Fk
r := ω ∈ U : Hk(ω) ≥ Hr(ω) .
Now, since Fkr is measurable for all r ∈ K , we obtain that Ek is measurable and so is Ω∗.
Furthermore, by construction, Ω∗ is in M`N . Since for all Ω ∈ M`
N we have that,
m∑j=1
∑s=1
| f j(ω + kΩs (ω))|2 ≤
m∑j=1
∑s=1
| f j(ω + k∗s(ω))|2 for almost all ω ∈ U,
taking the integral overU we getm∑
j=1
‖PVΩf j‖
2 =
∫U
m∑j=1
∑s=1
| f j(ω + kΩs (ω))|2 dω
≤
∫U
m∑j=1
∑s=1
| f j(ω + k∗s(ω))|2 dω
=
m∑j=1
‖PVΩ∗f j‖
2.
This shows that Ω∗ ∈ M`N is optimal over all Ω ∈ M`
N . We conclude that VΩ∗ ∈ T`N is a
solution for the data F .
Remark 4.3.6. Notice that if Ω∗N is the optimal multi-tile set for the class T `N for some data
F , then the approximation error EN(F , `) is given by
EN(F , `) =
∫Rd\Ω∗N
m∑j=1
| f j(ω)|2dω
=
∫CN\Ω
∗N
m∑j=1
| f j(ω)|2dω +
∫Rd\CN
m∑j=1
| f j(ω)|2dω.
4.3 Main results 63
Clearly EN(F , `) ≥ EN+1(F , `). So E(F , `) := limN→∞ EN(F , `) is somehow the optimalerror. Since F ⊆ L2(Rd) then the second integral goes to zero when N goes to infinite, forfunctions with good decay at infinite we will be close to the optimal error for convenientlylarge N.
Remark 4.3.7. In Proposition 4.3.4 we show, for an element of T `, how to construct aset of generators that gives a Riesz basis of translates in Zd. There are many ways toconstruct other sets of generators that give Riesz basis of integer translates. RecentlyGrepstad-Lev in [GL14] constructed a basis of exponentials for L2(Ω) when Ω ⊆ Rd is amulti-tile. Later on, Kolountzakis [Ko15] gave a simpler proof of this result in a slightlymore general form. Precisely they prove the following result.
Theorem 4.3.8 (Theorem 1,[Ko15]). Suppose Ω ⊆ Rd is bounded, measurable and multi-tiles Rd when translated by Zd at level `. Then there exist vectors a1, . . . , a` ∈ Rd such thatthe exponentials
e−2πi(a j+k)ω j = 1, . . . , `, k ∈ Zd
form a Riesz basis for L2(Ω).
From Theorem 4.3.8, we can obtain immediately a set of generators for VΩ. Let ϕ besuch that ϕ = χΩ. If a1, . . . , a` ∈ Rd are as in Theorem 4.3.8, then
VΩ = S (ϕ1, . . . , ϕ`), with ϕ j = ta jϕ, j = 1, . . . , `,
and the integer translates of ϕ1, . . . , ϕ` form a Riesz basis of VΩ.
In general, all the Riesz basis for VΩ can be described in the following way:
Let A = a js j,s ∈ [L2(U)]`×` be a measurable matrix, such that 0 < c1 ≤ λ(ω) ≤ c2
for every eigenvalue λ(ω) of A and for almost each ω ∈ U. Set k(ω) = (k1(ω), . . . , k`(ω))such that w + ks(ω) ∈ Ω. Define ϕ j such that ϕ j(ω + ks(w)) = a js(ω).
It is not difficult to see that ϕ1, . . . , ϕ` are measurable, VΩ = S (ϕ1, . . . , ϕ`) and theinteger translates of ϕ1, . . . , ϕ` form a Riesz basis of VΩ.
4.3.3 The discrete case
Consequently the Hilbert space we consider in this section is `2(Zd). We define the classof approximating subspaces in the following way:
Let N be an arbitrary finite set and Dσ : σ ∈ N be a partition of Zd, that is,
Zd =⋃σ∈N
Dσ,
where the union is disjoint.
For a ∈ `2(Zd), we denote aσ = 1Dσa, where 1Dσ
denotes the indicator of Dσ. GivenS ⊆ `2(Zd) a closed subspace we define
S σ := aσ : a ∈ S , for each σ ∈ N .
64 Subspaces with extra invariance nearest to observed data
We define the class of approximating subspaces by
D`N := S ⊆ `2(Zd) : S is a subspace, dim(S ) ≤ ` and S σ ⊆ S , ∀σ ∈ N. (4.18)
Note that S ∈ D`N
if and only if dim(S ) ≤ ` and S is the orthogonal sum of thesubspaces S σ i. e.,
S = ⊕σ∈NS σ.
For a given setA = a1, . . . , am ⊆ `2(Zd) consider for each σ ∈ N , the Gramian matrix
Gσ ∈ Cm×m of the data Aσ = aσ1 , . . . , a
σm, that is (Gσ)k,l = 〈aσk , a
σl , 〉, k, l = 1, . . . ,m.
Let λσ1 ≥ · · · ≥ λσm be the eigenvalues of the matrix Gσ and yσ1 , . . . , yσm the orthonormal
corresponding left eigenvectors.
Now setΛ := λσj : j = 1, . . . ,m, σ ∈ N
and collect in Λ` the ` first biggest eigenvalues of Λ, that is if λ ∈ Λ` then λ ≥ µ for allµ ∈ Λ \ Λ`. Write Λ` = λ1, . . . , λ`.
Now, we will proceed to define the sequence qs`s=1 ∈ `
2(Zd) :
Since λs = λσsjs
for some σs ∈ N and some js ∈ 1, . . . ,m, then λs is an eigenvalue ofGσs . Let yσs
jsbe the corresponding left eigenvector, yσs
js= (yσs
js(1), . . . , yσs
js(m)) ∈ Cm.
Then define, if λs ∈ Λ`, λs , 0
qs := (λs)−1/2m∑
k=1
yσsjs
(k)aσsk . (4.19)
If λs = 0 we define qs to be the zero sequence.
With this notation we can state the main theorem of this section:
Theorem 4.3.9. Let m, ` ∈ N and N a finite set. Assume that a set A = a1, . . . , am ⊆
`2(Zd) is given.
Then there exists S ∗ ∈ D`N
that satisfies
m∑j=1
‖a j − PS ∗a j‖2 ≤
m∑j=1
‖a j − PS a j‖2, ∀ S ∈ D`
N . (4.20)
Moreover, we have that
(1) S ∗ = spanq1, . . . , q` where q1, . . . , q` are defined in (4.19). Also, the vectorsq1, . . . , q` form a Parseval frame for S ∗.
(2) The error in the approximation is
E(A,N , `) =∑
λ∈Λ\Λ`
λ.
4.3 Main results 65
Proof. First, we observe that (4.20) is equivalently to,m∑
j=1
‖PS ∗a j‖2 ≥
m∑j=1
‖PS a j‖2, ∀ S ∈ D`
N .
Furthermore, if S ∈ D`N
then
m∑j=1
‖PS a j‖2 =
m∑j=1
‖∑σ∈N
PSσa j‖
2 =
m∑j=1
‖∑σ∈N
PSσaσj ‖
2 =∑σ∈N
m∑j=1
‖PSσaσj ‖
2,
where a j =∑σ∈N aσj .
In order to construct an optimal subspace S ∗ ∈ D`N
for the data A, we will find anoptimal subspace S σ for each σ ∈ N , of dimension at most ασ ∈ 1, . . . , ` for the dataAσ = aσ1 , . . . , a
σm. The existence of the optimal subspaces is provided by Theorem 4.2.3.
We need ∑σ∈N
dim(S σ) =∑σ∈N
ασ = dim(S ∗) ≤ `.
Thus if
Q :=
α = ασ : 0 ≤ ασ ≤ ` and∑σ∈N
ασ ≤ `
,then for each choice of α ∈ Q we will find optimal subspaces S α
σ : σ ∈ N and define
S α := ⊕σ∈NS ασ.
The candidate for S ∗ is the space S α which minimizes the expression (4.20) over allα ∈ Q.
Let β ∈ Q be the minimizer. Hence β satisfies,∑σ∈N
m∑j=1
‖PS ασaσj ‖
2 ≤∑σ∈N
m∑j=1
‖PS βσaσj ‖
2 ∀ α ∈ Q. (4.21)
Therefore, the subspace S ∗ := S β = ⊕σ∈NS βσ is the optimal subspace we need. It is
straightforward to see that S ∗ ∈ D`N
and that S ∗ is optimal.
Using Theorem 4.2.3, we obtain that for each α ∈ Q, the error of approximation for thedataAσ and the class of subspaces of dimension at most ασ is given by
E(Aσ, ασ) =
m∑s=ασ+1
λσs .
So the distance between the α-optimal subspace S α and the dataA is,
E(α) :=∑σ∈N
E(Aσ, ασ) =∑σ∈N
m∑s=ασ+1
λσs . (4.22)
66 Subspaces with extra invariance nearest to observed data
Let κ be the number of elements in N . We see that E(α) is minimum when the mκ − `eigenvalues used in (4.22) are the smallest from the set Λ = λσj : j = 1, . . . ,m, σ ∈ N.Therefore if we set Λ` ⊆ Λ the set of the ` biggest eigenvalues from Λ, the optimalβ = βσ ∈ Q satisfies that ⋃
σ∈N
λσ1 , . . . , λσβσ = Λ`.
Therefore,
E(A,N , `) =∑σ∈N
E(Aσ, βσ) =∑σ∈N
m∑j=1
‖aσj − PS βσaσj ‖
2 =∑σ∈N
m∑s=βσ+1
λσs =∑
λ∈Λ\Λ`
λ.
In order to construct the generators of S β it is enough to construct the generators ofeach S β
σ. Since S βσ are optimal subspaces for the data Aσ according with Theorem 4.2.3
the generators of S βσ are given by (4.19). That is the set qs : σs = σ is a Parseval frame
of S βσ.
Since the subspaces S βσ are mutually orthogonal, q1, . . . , q` is a set of Parseval frame
generators for the optimal space S ∗ = S β.
Remark 4.3.10. Even though the approximation problem for this discrete case is interest-ing by itself, that is related with the continuous case in L2(Rd) in a very interesting way.In fact, there is a reason to consider this particular class of subspaces for the discrete case.If a SIS V is M extra-invariant for some proper subgroup M of Rd containing Zd, then itsfiber spaces JV(ω) satisfies
JV(ω) =⊕σ∈N
JVσ(ω),
where JVσ(ω) = JV(ω)∩ `2(Bσ ∩ Zd). That is, they satisfies exactly the conditions that we
imposed on the class D`N
where the partition of Zd is Bσ ∩ Zd : σ ∈ N and Bσ and N
are defined as in (3.4).
The optimal subspace V∗ in Theorem 4.3.1 is the closest to the dataF over all subspacesV in the class V`
M. Using Lemma 1.4.4 and Lemma 1.4.5, it is not difficult to see thatalmost each fiber space JV∗(ω) ⊆ `2(Zd) of V∗ is the closest to the fibers of our dataτ(F )(ω) := τ f1(ω), . . . , τ fm(ω) over the particular class D`
Ndefined as above. (Clearly
this class is determine by the classV`M). So, the discrete result (Theorem 4.3.9) provides
a different proof of Theorem 4.3.1 using properties of range functions, without the needof Theorem 4.2.4.
4.4 Approximations in the context of LCA groups
We want to mention here, that the notion of shift invariant spaces still make sense inthe context of locally compact abelian groups. In that setting, the additive group Rd isreplaced with a general LCA group G and the additive subgroup Zd with a uniform latticeH ⊆ G, (see [CP10].)
Most of the result for the Euclidean case can be adapted, with certain difficulties, tothis abstract setting. In particular the notion of extra invariance ([ACP10]). In particular,after the completion of this thesis, the new results in chapter Chapter 2, were extended byPaternostro [Pat14].
We are working now in the extension of the approximation’s results to the setting ofLCA groups and its generalizations. This work is in progress and it is not going to be partof this thesis.
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Index
M−invariant space, 35, 38E(F ,S), 51, 52T `, 58T `
N , 60V`, 51V`
M, 55Vn, 54` multi-tile set, 59D`N
, 64e(F ,V), 53
analysis operator, 6
basisdual, 37orthonormal, 3Riezs, 4Schauder, 3
bundle, 53
dimension function, 11
Eckart-Young’s theorem, 50error, 50extra invariance, 36
fiber, 10space, 10
frame, 5canonical, 7coefficients, 8decomposition, 8dual, 7exact, 5generators, 24overcomplete, 7
Parseval, 5frame operator, 6Friedrichs angle, 20
Gramiandual, 8operator, 8
groupdimension, 37range, 37
invariance set, 35, 36
linear combination, 17
Minimum Subspace Approximation Prop-erty, 49
Moore-Penrose pseudoinverse, 27
optimal subspace, 49orthonormal generator, 54
Paley-Wiener space, 21, 35, 58Parseval Indentity, 4
range function, 10
sequenceBessel, 5frame, 5Riesz, 4
set of generators, 9minimal, 9, 17, 21
shift invariant space, 9finitely generated, 9Gramian, 12length, 9
74 INDEX
principal, 9singular value decomposition, 50spectrum, 8synthesis operator, 6
translation invariant space, 35
Wiener’s theorem, 35, 58